Algebra Concepts
An algebra foundations companion with concise notes, worked examples, and quick references for class, study, and review.
Course Map
Algebra Concepts Modules
Four modules organized by lettered homework topic.
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Module 1: Linear Equations, Inequalities, and Functions
Solve and model with linear equations and inequalities, graph lines, and build the function language needed for nonlinear and piecewise functions.
Mod 1 A
Solving Linear Equations
Solve linear equations from one-step forms through equations with distribution, fractions, and special solution cases.
Quick reference
- Use inverse operations to isolate the variable while keeping both sides balanced.
- Distribute and combine like terms before isolating the variable.
- Clear fractions by multiplying every term by a common denominator.
- A true statement after the variable disappears means all real numbers; a false statement means no solution.
- Check a proposed solution in the original equation.
Example 1. Solve \(-3y+12=-3(2y+5)\).
Answer: \(y=-9\).
Example 2. Solve \(\dfrac{5x}{6}-7=\dfrac{x}{3}+3\).
Answer: \(x=20\).
Example 3. Solve \(-5(4x-5)=5x\).
Answer: \(x=1\).
Example 4. Solve \(4(2n-3)+5=3(n+1)\).
Answer: \(n=2\).
Example 5. Solve and classify \(2(3x+4)=6x+8\).
Answer: All real numbers.
Example 6. Solve and classify \(3(x-2)=3x+5\).
Answer: No solution, \(\varnothing\).
Mod 1 B
Translating Equations and Word Problems
Translate verbal statements into equations and use those equations to solve number and applied problems.
Quick reference
- Choose a variable and state what it represents before writing the equation.
- Words such as sum, difference, product, quotient, more than, and less than signal operations.
- “Less than” reverses the written order: five less than \(x\) is \(x-5\).
- Translate the relationship first, solve second, and answer the question with units.
- Check whether the result is reasonable in the original context.
Example 1. The sum of twice a number and \(7\) equals the sum of the number and \(19\). Write an equation and find the number.
Answer: \(2x+7=x+19\), so the number is \(12\).
Example 2. Four times a number minus \(9\) equals three times the number plus \(4\). Find the number.
Answer: \(4x-9=3x+4\), so \(x=13\).
Example 3. Twice the difference of a number and \(5\) is \(18\). Find the number.
Answer: \(2(x-5)=18\), so \(x=14\).
Example 4. Two consecutive integers have a sum of \(51\). Find them.
Answer: \(x+(x+1)=51\). The integers are \(25\) and \(26\).
Example 5. A streaming service charges \(\$12\) plus \(\$3\) per movie. A bill is \(\$33\). How many movies were rented?
Answer: \(12+3m=33\), so \(7\) movies were rented.
Mod 1 C
Equations With More Than One Variable
Rearrange formulas for a requested variable and use common formulas in applications.
Quick reference
- A literal equation contains two or more variables. Solve for the requested variable as if the other variables were known numbers.
- Undo addition and subtraction before multiplication and division when isolating a variable.
- Factor out the requested variable when it appears in more than one term.
- Common applications use \(A=lw\), \(d=rt\), and perimeter formulas.
- Keep units attached when substituting into a formula.
Example 1. Solve \(V=APK\) for \(A\).
Answer: \(A=\dfrac{V}{PK}\).
Example 2. Solve \(2x+y=7\) for \(y\).
Answer: \(y=7-2x\).
Example 3. Solve \(P=a+b+c\) for \(c\).
Answer: \(c=P-a-b\).
Example 4. Solve \(y=mx+b\) for \(x\), assuming \(m\ne0\).
Answer: \(x=\dfrac{y-b}{m}\).
Example 5. A rectangular sign has area \(3763\) square feet and width \(53\) feet. Find its height.
Answer: Using \(A=wh\), \(h=\dfrac{3763}{53}=71\) feet.
Example 6. A ferry travels at \(59\) miles per hour for \(3\dfrac12\) hours. How far does it travel?
Answer: \(d=rt=59(3.5)=206.5\) miles.
Mod 1 D
Absolute Value Equations
Solve absolute value equations by separating distance statements into cases.
Quick reference
- Absolute value measures distance from zero, so \(|u|=k\) with \(k>0\) gives \(u=k\) or \(u=-k\).
- Isolate the absolute value expression before writing the two cases.
- If \(|u|=0\), then \(u=0\) gives one solution.
- An absolute value cannot equal a negative number, so \(|u|=-k\) has no solution when \(k>0\).
- Check both candidates in the original equation.
Example 1. Solve \(|2x-11|=17\).
Answer: \(x=14\) or \(x=-3\).
Example 2. Solve \(\left|\dfrac{x}{4}+9\right|=9\).
Answer: \(x=0\) or \(x=-72\).
Example 3. Solve \(|8z|=0\).
Answer: \(z=0\).
Example 4. Solve \(3|x+2|-4=17\).
Answer: \(x=5\) or \(x=-9\).
Example 5. Solve \(5-|2y-1|=9\).
Answer: No solution, \(\varnothing\).
Mod 1 E
Solving Linear Inequalities
Solve linear inequalities and express solutions with inequality, interval, and number-line notation.
Quick reference
- Solve a linear inequality like an equation, but reverse the inequality sign when multiplying or dividing by a negative number.
- Use an open endpoint for \(<\) or \(>\), and a closed endpoint for \(\le\) or \(\ge\).
- Parentheses mark excluded interval endpoints; brackets mark included endpoints.
- Infinity always uses a parenthesis.
- Translate verbal conditions into inequalities before solving, and keep units attached in applications.
Example 1. Solve \(x-4\ge-4\). Give interval notation and a number-line graph.
Answer: \(x\ge0\), or \([0,\infty)\).
Closed point at zero with shading to the right
Example 2. Solve \(5x-6>4x-4\). Give interval notation and a number-line graph.
Answer: \(x>2\), or \((2,\infty)\).
Open point at two with shading to the right
Example 3. Solve \(-3(2x-1)\le15\).
Answer: \(x\ge-2\), or \([-2,\infty)\).
Closed point at negative two with shading to the right
Example 4. Solve \(\dfrac{x-1}{3}<\dfrac{x+5}{2}\).
Answer: \(x>-17\), or \((-17,\infty)\).
Example 5. Seven more than twice a number is less than \(-13\). Find all numbers that make the statement true.
Answer: \(2x+7<-13\), so \(x<-10\), or \(( -\infty,-10)\).
Example 6. A rectangle has perimeter no greater than \(80\) centimeters and width \(15\) centimeters. Find its maximum length.
Answer: \(2L+2(15)\le80\), so \(L\le25\). The maximum length is \(25\) centimeters.
Mod 1 F
Compound Inequalities
Solve intersections joined by “and” and unions joined by “or.”
Quick reference
- An “and” compound inequality requires both conditions, so keep the overlap of the solution sets.
- An “or” compound inequality accepts either condition, so combine both solution sets.
- A three-part inequality can be solved by performing the same operation on all three parts.
- Use \(\cup\) to join separated intervals.
- Some “and” statements have no overlap; some “or” statements cover all real numbers.
Example 1. Solve \(x+10\ge5\) and \(2x-4\ge4\).
Answer: \(x\ge4\), or \([4,\infty)\).
Closed point at four with shading to the right
Example 2. Solve \(-3x<-3\) and \(x-4<6\).
Answer: \(1<x<10\), or \((1,10)\).
Open points at one and ten with shading between
Example 3. Solve \(-7\le2x+1<9\).
Answer: \(-4\le x<4\), or \([-4,4)\).
Example 4. Solve \(3x-2\le-8\) or \(2x+5>11\).
Answer: \(x\le-2\) or \(x>3\), or \(( -\infty,-2]\cup(3,\infty)\).
Closed point at negative two shading left and open point at three shading right
Example 5. Solve \(x>5\) and \(x\le2\).
Answer: No solution, \(\varnothing\).
Mod 1 G
Absolute Value Inequalities
Translate absolute value inequalities into compound inequalities and graph their solution sets.
Quick reference
- For \(|u|<k\), values lie inside: \(-k<u<k\). The same inside pattern works with \(\le\).
- For \(|u|>k\), values lie outside: \(u<-k\) or \(u>k\). The same outside pattern works with \(\ge\).
- Isolate the absolute value before applying the inside or outside pattern.
- When \(k<0\), compare the nonnegative absolute value with a negative number before doing algebra.
- Match strict inequalities with open endpoints and inclusive inequalities with closed endpoints.
Example 1. Solve \(|x+5|<4\).
Answer: \(-9<x<-1\), or \((-9,-1)\).
Open points at negative nine and negative one with shading between
Example 2. Solve \(|2x+5|\le9\).
Answer: \(-7\le x\le2\), or \([-7,2]\).
Closed points at negative seven and two with shading between
Example 3. Solve \(|3x-1|>8\).
Answer: \(x<-\dfrac73\) or \(x>3\).
Example 4. Solve \(2|x-4|+1\ge11\).
Answer: \(x\le-1\) or \(x\ge9\), or \(( -\infty,-1]\cup[9,\infty)\).
Example 5. Solve \(|x-2|<-3\).
Answer: No solution, \(\varnothing\).
Example 6. Solve \(|x-2|\ge-3\).
Answer: All real numbers, \(( -\infty,\infty)\).
Mod 1 H
Graphing Linear Equations by Plotting Points
Create ordered-pair tables and graph linear equations from calculated points.
Quick reference
- Choose convenient input values, substitute them into the equation, and record the ordered pairs.
- Two distinct points determine a line; a third point helps check the work.
- Plot coordinates as \((x,y)\), then draw one straight line through the points.
- For \(y=mx+b\), the y-intercept \((0,b)\) is often a convenient first point.
- For equations not solved for \(y\), choose x-values that make the corresponding y-values easy to calculate.
Example 1. Complete a table for \(y=-2x\) at \(x=-1,0,1\), then graph the line.
Answer: The points are \((-1,2),(0,0),(1,-2)\).
| \(x\) | \(-1\) | \(0\) | \(1\) |
|---|---|---|---|
| \(y\) | \(2\) | \(0\) | \(-2\) |
A decreasing line passes through negative one comma two, zero comma zero, and one comma negative two
Example 2. Complete a table for \(y=3x+6\) at \(x=-2,-1,0\), then graph the line.
Answer: The points are \((-2,0),(-1,3),(0,6)\).
An increasing line passes through negative two comma zero, negative one comma three, and zero comma six
Example 3. Find three points on \(2x+y=4\) and graph the line.
Answer: One convenient set is \((0,4),(1,2),(2,0)\).
A decreasing line passes through zero comma four, one comma two, and two comma zero
Example 4. Graph \(y=\dfrac12x+1\) by plotting three points.
Answer: One convenient set is \((-2,0),(0,1),(2,2)\).
An increasing line passes through negative two comma zero, zero comma one, and two comma two
Mod 1 I
Graphing Linear Equations by Plotting Intercepts
Find x- and y-intercepts and use them to graph linear equations.
Quick reference
- To find the x-intercept, set \(y=0\) and solve for \(x\).
- To find the y-intercept, set \(x=0\) and solve for \(y\).
- Plot both intercepts and draw the line through them.
- A line through the origin has the same x- and y-intercept, so find another point.
- Horizontal and vertical lines may have only one type of intercept.
Example 1. Find the intercepts and graph \(x-y=3\).
Answer: x-intercept: \((3,0)\). y-intercept: \((0,-3)\).
An increasing line crosses the axes at three comma zero and zero comma negative three
Example 2. Find the intercepts and graph \(-x+3y=9\).
Answer: x-intercept: \((-9,0)\). y-intercept: \((0,3)\).
An increasing line crosses the axes at negative nine comma zero and zero comma three
Example 3. Find the intercepts and graph \(6x-2y=-6\).
Answer: x-intercept: \((-1,0)\). y-intercept: \((0,3)\).
An increasing line crosses the axes at negative one comma zero and zero comma three
Example 4. Graph \(4x+2y=0\) using its intercept and one additional point.
Answer: Both intercepts are \((0,0)\). Another point is \((1,-2)\).
A decreasing line passes through the origin and one comma negative two
Mod 1 J
Slope
Find slope from ordered pairs and linear equations.
Quick reference
- Slope is \(m=\dfrac{y_2-y_1}{x_2-x_1}\), or rise divided by run.
- A horizontal line has slope \(0\).
- A vertical line has undefined slope because its run is \(0\).
- In \(y=mx+b\), the coefficient \(m\) is the slope.
- For an equation in standard form, solve for \(y\) to identify the slope.
Example 1. Find the slope through \((10,-5)\) and \((8,5)\).
Answer: \(m=\dfrac{5-(-5)}{8-10}=-5\).
Example 2. Find the slope through \((-8,1)\) and \((-8,10)\).
Answer: The slope is undefined.
Example 3. Find the slope through \((-3,6)\) and \((4,6)\).
Answer: \(m=0\).
Example 4. Find the slope of \(y=4x+7\).
Answer: The slope is \(4\).
Example 5. Find the slope of \(3x-2y=6\).
Answer: Solving for \(y\) gives \(y=\dfrac32x-3\), so the slope is \(\dfrac32\).
Mod 1 K
Writing Equations of Lines
Write slope-intercept equations from slopes, points, and intercepts.
Quick reference
- Slope-intercept form is \(y=mx+b\).
- Point-slope form is \(y-y_1=m(x-x_1)\).
- Find slope first when two points are given, then substitute one point.
- If the y-intercept is given, place it directly into \(y=mx+b\).
- Simplify coefficients and constants before writing the final equation.
Example 1. Write the line with slope \(-\dfrac29\) through \((0,8)\) in slope-intercept form.
Answer: \(y=-\dfrac29x+8\).
Example 2. Write the line through \((6,3)\) and \((3,6)\).
Answer: \(y=-x+9\).
Example 3. Write the line with slope \(3\) through \((-4,12)\).
Answer: \(y=3x+24\).
Example 4. Write the line with slope \(-4\) and y-intercept \((0,4)\).
Answer: \(y=-4x+4\).
Example 5. Write the line through \((0,0)\) and \((4,7)\).
Answer: \(y=\dfrac74x\).
Example 6. Write the line with slope \(-\dfrac56\) through \((-1,-7)\).
Answer: \(y=-\dfrac56x-\dfrac{47}{6}\).
Mod 1 L
Functions, Domain, and Range
Identify functions and determine domain and range from relations and equations.
Quick reference
- A relation is a function when every input has exactly one output.
- Domain is the set of inputs; range is the set of outputs. Do not repeat values in a set.
- Repeated outputs are allowed, but a repeated input cannot have different outputs.
- An equation describes a function when each x-value determines exactly one y-value.
- A vertical equation such as \(x=c\) does not describe \(y\) as a function of \(x\).
Example 1. Find the domain and range of \(\{(8,7),(1,5),(-8,9),(9,-8)\}\).
Answer: Domain: \(\{-8,1,8,9\}\). Range: \(\{-8,5,7,9\}\).
Example 2. Find the domain and range of \(\{(10,5),(5,5),(2,5)\}\).
Answer: Domain: \(\{2,5,10\}\). Range: \(\{5\}\).
Example 3. Is \(\{(7,-8),(-2,-5),(8,8),(-3,9)\}\) a function?
Answer: Yes. Every input occurs only once.
Example 4. Is \(\{(1,4),(2,6),(1,9)\}\) a function?
Answer: No. Input \(1\) has two outputs.
Example 5. Does \(y=-2x-4\) describe a function?
Answer: Yes. Every x-value gives exactly one y-value.
Example 6. Does \(x=-4\) describe \(y\) as a function of \(x\)?
Answer: No. The single input \(x=-4\) is paired with many y-values.
Mod 1 M
Function Notation
Evaluate functions at specified inputs using function notation.
Quick reference
- \(f(a)\) means substitute \(a\) for every \(x\) in the rule.
- Function notation does not mean multiplication.
- Use parentheses carefully when substituting negative numbers.
- Evaluate each requested input separately and simplify the resulting number.
- The same substitution process works for linear, quadratic, and other polynomial rules.
Example 1. For \(f(x)=-5x-5\), find \(f(-1)\), \(f(0)\), and \(f(3)\).
Answer: \(f(-1)=0\), \(f(0)=-5\), and \(f(3)=-20\).
Example 2. For \(g(x)=x^2+5\), find \(g(-3)\), \(g(0)\), and \(g(3)\).
Answer: \(g(-3)=14\), \(g(0)=5\), and \(g(3)=14\).
Example 3. For \(h(x)=2x^3\), find \(h(-2)\), \(h(0)\), and \(h(3)\).
Answer: \(h(-2)=-16\), \(h(0)=0\), and \(h(3)=54\).
Example 4. For \(r(x)=-7x\), find \(r(-2)\), \(r(0)\), and \(r(2)\).
Answer: \(r(-2)=14\), \(r(0)=0\), and \(r(2)=-14\).
Example 5. For \(q(x)=6x^2+5\), find \(q(-3)\), \(q(0)\), and \(q(1)\).
Answer: \(q(-3)=59\), \(q(0)=5\), and \(q(1)=11\).
Mod 1 N
Equations of Lines in Function Notation
Write linear equations using function notation.
Quick reference
- Replace \(y\) with \(f(x)\) to write a line as a function.
- A line with slope \(m\) and y-intercept \((0,b)\) is \(f(x)=mx+b\).
- Use point-slope form to find the rule when a slope and non-intercept point are given.
- When two points are given, calculate the slope before finding the y-intercept.
- A slope of \(0\) produces a constant function \(f(x)=b\).
Example 1. Write the line with slope \(-2\) and y-intercept \((0,9)\) using function notation.
Answer: \(f(x)=-2x+9\).
Example 2. Write the line with slope \(4\) and y-intercept \((0,-\dfrac29)\) using function notation.
Answer: \(f(x)=4x-\dfrac29\).
Example 3. Write the line with slope \(3\) through \((2,1)\) using function notation.
Answer: \(f(x)=3x-5\).
Example 4. Write the function through \((-1,5)\) and \((3,-3)\).
Answer: \(f(x)=-2x+3\).
Example 5. Write the line with slope \(-9\) through \((7,8)\) using function notation.
Answer: \(f(x)=-9x+71\).
Example 6. Write the line with slope \(0\) through \((-1,-1)\) using function notation.
Answer: \(f(x)=-1\).
Mod 1 O
Nonlinear Functions
Graph basic quadratic, absolute-value, and square-root functions.
Quick reference
- Quadratic, absolute-value, and square-root rules produce nonlinear graphs.
- A quadratic graph is a parabola; an absolute value graph is V-shaped.
- Square-root graphs have an endpoint and extend in one direction.
- Read horizontal and vertical shifts directly from the function before selecting or sketching its graph.
- Use several accurate points to confirm the graph's location and orientation.
Example 1. Graph \(f(x)=x^2-1\).
Answer: The graph is an upward-opening parabola with vertex \((0,-1)\).
An upward-opening parabola has vertex at zero comma negative one
Example 2. Graph \(g(x)=|x|-2\).
Answer: The graph is V-shaped with vertex \((0,-2)\).
A V-shaped graph has vertex at zero comma negative two
Example 3. Graph \(h(x)=\sqrt{x+4}\).
Answer: The graph begins at \((-4,0)\) and extends right.
A square-root curve begins at negative four comma zero and rises to the right
Example 4. Graph \(p(x)=|x|-3\).
Answer: The graph is V-shaped with vertex \((0,-3)\).
A V-shaped graph has vertex at zero comma negative three
Example 5. Graph \(q(x)=-(x+5)^2\).
Answer: The graph is a downward-opening parabola with vertex \((-5,0)\).
A downward-opening parabola has vertex at negative five comma zero
Mod 1 P
Piecewise Functions
Graph piecewise functions and determine domain and range.
Quick reference
- A piecewise function uses different rules on different parts of its domain.
- Graph each rule only over the interval named by its condition.
- Use a closed endpoint when the boundary value is included and an open endpoint when it is excluded.
- At a breakpoint, only one output can be included if the relation is a function.
- Read domain and range from every piece, including gaps and endpoints.
Example 1. Graph \(f(x)=\begin{cases}3x,&x<0\\x+4,&x\ge0\end{cases}\).
Answer: The first piece approaches an open point at \((0,0)\); the second begins with a closed point at \((0,4)\).
The left line approaches an open point at zero comma zero and the right line begins at a closed point at zero comma four
Example 2. Graph \(g(x)=\begin{cases}-x,&x\le-2\\2x+1,&x>-2\end{cases}\).
Answer: The first piece ends with a closed point at \((-2,2)\); the second approaches an open point at \((-2,-3)\).
The left line ends at a closed point negative two comma two and the right line begins at an open point negative two comma negative three
Example 3. Graph \(h(x)=\begin{cases}-3x,&x\le0\\3x+3,&x>0\end{cases}\), then state its domain and range.
Answer: Domain: \(( -\infty,\infty)\). Range: \([0,\infty)\).
The left piece includes zero comma zero and the right piece approaches an open point at zero comma three
Example 4. Graph \(p(x)=\begin{cases}2x-4,&x\le-1\\-x+3,&x>-1\end{cases}\), then state its domain and range.
Answer: Domain: \(( -\infty,\infty)\). Range: \(( -\infty,4)\).
The left line ends at a closed point negative one comma negative six and the right line approaches an open point negative one comma four
Example 5. Graph \(q(x)=\begin{cases}x+5,&x<-1\\-2x+3,&x\ge-1\end{cases}\), then state its domain and range.
Answer: Domain: \(( -\infty,\infty)\). Range: \(( -\infty,5]\).
The left line approaches an open point negative one comma four and the right line begins at a closed point negative one comma five
Module 2: Exponents, Polynomials, and Quadratic Foundations
Use exponent rules, operate on and divide polynomials, factor completely, and solve foundational quadratic equations and applications.
Mod 2 A
Exponent Rules
Simplify expressions using the product, quotient, and power rules.
Quick reference
- For the same base, multiply by adding exponents: \(a^m a^n=a^{m+n}\).
- Divide same bases by subtracting exponents: \(a^m/a^n=a^{m-n}\), for \(a\ne0\).
- A power raised to a power multiplies exponents: \((a^m)^n=a^{mn}\).
- Apply a power to every factor in a product or quotient.
- Simplify numerical coefficients and write the result with positive exponents.
Example 1. Simplify \((8y^3)(7y)\).
Answer: \(56y^4\).
Example 2. Simplify \((-9xy^4)(4x^4y^5)\).
Answer: \(-36x^5y^9\).
Example 3. Simplify \((4c^7)^3\).
Answer: \(64c^{21}\).
Example 4. Simplify \(\left(\dfrac{5x^5z^3}{y^2}\right)^2\).
Answer: \(\dfrac{25x^{10}z^6}{y^4}\).
Example 5. Simplify \(\dfrac{p^3q^7}{p^2q^2}\).
Answer: \(pq^5\).
Example 6. Simplify \((-3xyz^4)^2\).
Answer: \(9x^2y^2z^8\).
Example 7. Simplify \(\left(\dfrac{4u^2}{8v^4}\right)^5\).
Answer: \(\dfrac{u^{10}}{32v^{20}}\).
Mod 2 B
Polynomial Introduction
Find polynomial degree and classify polynomials by their number of terms.
Quick reference
- A polynomial is made of terms with numerical coefficients and nonnegative whole-number exponents.
- The degree of a one-variable polynomial is its greatest exponent.
- The degree is the greatest term degree; for multiple variables, add the exponents within a term.
- A monomial has one term, a binomial two, and a trinomial three.
- A polynomial with more than three terms is classified as none of these.
Example 1. Find the degree and classify \(x^3+6\).
Answer: Degree \(3\); binomial.
Example 2. Find the degree and classify \(6m^4-5m^3+7m-8\).
Answer: Degree \(4\); neither monomial, binomial, nor trinomial.
Example 3. Find the degree of \(4zx+2z^3x^2+3zx^3\).
Answer: Degree \(5\), from \(2z^3x^2\).
Example 4. Find the degree and classify \(6-4x^3\).
Answer: Degree \(3\); binomial.
Example 5. Find the degree and classify \(3xy^4-3\).
Answer: Degree \(5\); binomial.
Example 6. Find the degree and classify \(2a^8+4a-5\).
Answer: Degree \(8\); trinomial.
Mod 2 C
Polynomial Operations
Add, subtract, and multiply polynomials, including common binomial products.
Quick reference
- Add and subtract polynomials by combining like terms.
- When subtracting a polynomial, distribute the negative sign to every term.
- Multiply each term by every term in the other polynomial.
- For a binomial square, multiply the binomial by itself and combine like terms.
- Write the simplified result in standard form.
Example 1. Add \((-2x+10)+(7x^2+2x+12)\).
Answer: \(7x^2+22\).
Example 2. Subtract \((7y^2+8y-5)-(-2y+2)\).
Answer: \(7y^2+10y-7\).
Example 3. Multiply \(-2a^2(5a^2-2a+5)\).
Answer: \(-10a^4+4a^3-10a^2\).
Example 4. Multiply \((x+2)(x+7)\).
Answer: \(x^2+9x+14\).
Example 5. Expand \((5y+5)^2\).
Answer: \(25y^2+50y+25\).
Example 6. Multiply \((x-2)(x^2-7x+3)\).
Answer: \(x^3-9x^2+17x-6\).
Example 7. Multiply \((x+4)(x^3-3x+5)\).
Answer: \(x^4+4x^3-3x^2-7x+20\).
Mod 2 D
Scientific Notation
Convert between standard and scientific notation for very large and very small numbers.
Quick reference
- Scientific notation has the form \(a\times10^n\), where \(1\le|a|<10\).
- A positive exponent moves the decimal right when returning to standard notation.
- A negative exponent moves the decimal left when returning to standard notation.
- Count decimal-place moves carefully when converting a standard number to scientific notation.
- Include placeholder zeros when writing a small number in standard notation.
Example 1. Write \(43{,}000\) in scientific notation.
Answer: \(4.3\times10^4\).
Example 2. Write \(0.00000161\) in scientific notation.
Answer: \(1.61\times10^{-6}\).
Example 3. Write \(1.667\times10^{-7}\) in standard notation.
Answer: \(0.0000001667\).
Example 4. Write \(6.1\times10^{-2}\) in standard notation.
Answer: \(0.061\).
Example 5. Write \(2.802\times10^4\) in standard notation.
Answer: \(28{,}020\).
Example 6. A star converts \(3.0\times10^7\) tons of hydrogen each second. Write that amount in standard notation.
Answer: \(30{,}000{,}000\) tons.
Mod 2 E
Polynomial Division
Divide polynomials by monomials and binomials using termwise division and long division.
Quick reference
- Divide every numerator term when the divisor is a monomial.
- For long division, arrange both polynomials in descending powers and include zero placeholders for missing terms.
- Divide leading terms, multiply, subtract, and bring down the next term.
- Write a nonzero remainder over the divisor.
- Check with \(\text{dividend}=(\text{divisor})(\text{quotient})+\text{remainder}\).
Example 1. Divide \(\dfrac{9p^4+15p^3}{3p}\).
Answer: \(3p^3+5p^2\).
Example 2. Divide \(\dfrac{-4x^7+12x^5-8}{4x^2}\).
Answer: \(-x^5+3x^3-\dfrac{2}{x^2}\).
Example 3. Divide \(\dfrac{x^2+9x+20}{x+5}\).
Answer: \(x+4\).
Example 4. Divide \(\dfrac{6x^2-16x+2}{x-3}\).
Answer: \(6x+2+\dfrac{8}{x-3}\).
Example 5. Divide \(\dfrac{2x^3+3x-5}{x+2}\).
Answer: \(2x^2-4x+11-\dfrac{27}{x+2}\).
Example 6. Check the division \(x^3-1=(x-1)(x^2+x+1)\).
Answer: Multiplying the divisor and quotient gives \(x^3-1\), so the remainder is \(0\).
Mod 2 F
Synthetic Division
Use synthetic division for divisors of the form x minus c.
Quick reference
- Synthetic division applies when the divisor is \(x-c\). Use \(c\) in the synthetic setup.
- For \(x+a\), the synthetic value is \(-a\).
- List every coefficient in descending order, using zero for missing powers.
- Bring down, multiply, and add repeatedly.
- The final value is the remainder; the other entries are quotient coefficients.
Example 1. Use synthetic division to divide \(x^2+3x-28\) by \(x+7\).
Answer: \(x-4\).
Example 2. Use synthetic division to divide \(x^3-9x^2-6x+5\) by \(x-3\).
Answer: \(x^2-6x-24-\dfrac{67}{x-3}\).
Example 3. Use synthetic division to divide \(17x^2-30\) by \(x+3\).
Answer: \(17x-51+\dfrac{123}{x+3}\).
Example 4. Use synthetic division to divide \(4x^3+2x^2-4x+1\) by \(x-\dfrac12\).
Answer: \(4x^2+4x-2\).
Example 5. Use synthetic division to divide \(3x^3+15x^2-4x-23\) by \(x+5\).
Answer: \(3x^2-4-\dfrac{3}{x+5}\).
Mod 2 G
The Remainder Theorem
Use the Remainder Theorem to evaluate polynomials at specified values.
Quick reference
- When \(P(x)\) is divided by \(x-c\), the remainder is \(P(c)\).
- Synthetic substitution computes \(P(c)\) efficiently.
- Use every coefficient in descending order, including zero placeholders for missing powers.
- Bring down, multiply by \(c\), and add to obtain the next entry.
- Direct substitution and synthetic division should give the same value.
Example 1. For \(P(x)=x^3+5x^2-4x+4\), find \(P(4)\).
Answer: \(P(4)=132\).
Example 2. For \(P(x)=6x^3-6x^2-5x+2\), find \(P(-3)\).
Answer: \(P(-3)=-199\).
Example 3. For \(P(x)=2x^4-4x^2+2\), find \(P(-1)\).
Answer: \(P(-1)=0\).
Example 4. For \(P(x)=x^5-x^4-x^3-5\), find \(P\!\left(-\dfrac12\right)\).
Answer: \(P\!\left(-\dfrac12\right)=-\dfrac{159}{32}\).
Example 5. For \(P(x)=3x^4-2x+7\), find \(P(2)\).
Answer: \(P(2)=51\).
Mod 2 H
Factoring
Factor polynomials completely using GCF, grouping, trinomial patterns, and special products.
Quick reference
- Always factor out the greatest common factor first.
- Four-term polynomials may factor by grouping.
- For \(x^2+bx+c\), find two numbers whose product is \(c\) and sum is \(b\).
- A difference of squares factors as \(a^2-b^2=(a-b)(a+b)\).
- If no integer factorization exists, classify the polynomial as prime.
Example 1. Factor by grouping: \(x^3+7x^2+9x+63\).
Answer: \((x+7)(x^2+9)\).
Example 2. Factor completely: \(2x^3-12x^2+16x\).
Answer: \(2x(x-2)(x-4)\).
Example 3. Factor by grouping: \(15x^3-6x^2+10x-4\).
Answer: \((5x-2)(3x^2+2)\).
Example 4. Factor \(x^2-x-72\).
Answer: \((x-9)(x+8)\).
Example 5. Factor \(6x^2+x-2\).
Answer: \((3x+2)(2x-1)\).
Example 6. Factor \(5m^2+11m+15\), or state that it is prime.
Answer: The polynomial is prime.
Example 7. Factor \(4x^2-81\).
Answer: \((2x-9)(2x+9)\).
Example 8. Factor \(9x^2+4\), or state that it is prime.
Answer: The polynomial is prime.
Mod 2 I
Solving Quadratic Equations by Factoring
Use the zero-product property to solve factorable quadratic equations.
Quick reference
- Write the equation with zero on one side before factoring.
- Factor completely, then set each factor equal to zero.
- The zero-product property applies only when a product equals zero.
- A repeated factor gives a repeated solution that should be listed once.
- Check solutions in the original equation when expansion or rearrangement was required.
Example 1. Solve \((x-1)(x+1)=0\).
Answer: \(x=-1,1\).
Example 2. Solve \(x(x+3)=0\).
Answer: \(x=-3,0\).
Example 3. Solve \((5x-8)(7x+6)=0\).
Answer: \(x=\dfrac85,-\dfrac67\).
Example 4. Solve \(x^2-12x+32=0\).
Answer: \(x=4,8\).
Example 5. Solve \(x^2-2x=24\).
Answer: \(x=-4,6\).
Example 6. Solve \(10x^2-21x-10=0\).
Answer: \(x=-\dfrac25,\dfrac52\).
Example 7. Solve \(4x^2-12x+9=0\).
Answer: \(x=\dfrac32\), a repeated solution.
Mod 2 J
Applications of Quadratic Equations
Translate applications into quadratic equations and choose solutions that fit the context.
Quick reference
- Define the variable and write the quadratic model before solving.
- Projectile height in feet often uses \(h(t)=-16t^2+v_0t+h_0\).
- Area applications use products of related dimensions.
- Set the model equal to the requested output, move all terms to one side, and factor when possible.
- Reject negative time, length, or quantity values that do not make sense.
Example 1. An object is thrown from a \(96\)-foot building with initial velocity \(80\) ft/s. Its height is \(h(t)=-16t^2+80t+96\). When does it hit the ground?
Answer: \(t=6\) seconds; reject \(t=-1\).
Example 2. A rectangle is \(5\) centimeters longer than its width and has area \(84\) square centimeters. Find its dimensions.
Answer: \(w(w+5)=84\). The width is \(7\) cm and the length is \(12\) cm.
Example 3. The sides of a square are increased by \(5\) inches, producing an area of \(169\) square inches. Find the original side length.
Answer: \((x+5)^2=169\). The original side is \(8\) inches.
Example 4. An object is dropped from \(484\) feet, so \(h(t)=-16t^2+484\). When does it reach the ground?
Answer: \(t=5.5\) seconds.
Example 5. Manufacturing cost is \(C(x)=x^2+15x+55\). How many units give a cost of \(\$10{,}505\)?
Answer: Solve \(x^2+15x+55=10505\): \(x=95\) units; reject \(x=-110\).
Module 3: Rational Expressions, Radicals, and Complex Numbers
Work with rational expressions and equations, build radical and rational-exponent skills, apply coordinate formulas, and operate with complex numbers.
Mod 3 A
Domain of Rational Expressions
Find domain restrictions by identifying values that make a rational denominator zero.
Quick reference
- A rational expression is undefined wherever its denominator equals zero.
- Set the denominator equal to zero and solve to find excluded values.
- Factor a polynomial denominator completely so every restriction is visible.
- Restrictions come from the original denominator, even if a factor later cancels.
- State the domain with exclusions, set-builder notation, or interval notation.
Example 1. Find the domain of \(f(x)=\dfrac{5x-6}{7}\).
Answer: All real numbers, \(( -\infty,\infty)\).
Example 2. Find the domain of \(f(x)=\dfrac{5x}{5x+8}\).
Answer: \(x\ne-\dfrac85\).
Example 3. Find the domain of \(R(x)=\dfrac{4+6x}{x^3+2x^2-3x}\).
Answer: The denominator is \(x(x+3)(x-1)\), so \(x\ne-3,0,1\).
Example 4. Find the domain of \(C(x)=\dfrac{x-3}{x^2-16}\).
Answer: \(x\ne-4,4\).
Example 5. Find the domain of \(g(t)=\dfrac{t+2}{t^2+9}\).
Answer: All real numbers because \(t^2+9\) is never zero for real \(t\).
Mod 3 B
Simplifying Rational Expressions
Factor and simplify rational expressions while preserving restrictions from the original expression.
Quick reference
- Factor numerators and denominators before canceling.
- Cancel common factors, not individual terms joined by addition or subtraction.
- Opposite binomials differ by a factor of \(-1\): \(a-b=-(b-a)\).
- Keep every excluded value from the original denominator.
- A rational expression is simplified when numerator and denominator share no nonconstant factor.
Example 1. Simplify \(\dfrac{x+6}{6+x}\).
Answer: \(1\), with \(x\ne-6\).
Example 2. Simplify \(\dfrac{x-8}{8-x}\).
Answer: \(-1\), with \(x\ne8\).
Example 3. Simplify \(\dfrac{5}{15x+75}\).
Answer: \(\dfrac{1}{3(x+5)}\), with \(x\ne-5\).
Example 4. Simplify \(\dfrac{-4x+4y}{x-y}\).
Answer: \(-4\), with \(x\ne y\).
Example 5. Simplify \(\dfrac{3x^2-x-4}{3x-4}\).
Answer: \(x+1\), with \(x\ne\dfrac43\).
Example 6. Simplify \(\dfrac{x^3+7x^2}{x^2+3x-28}\).
Answer: \(\dfrac{x^2}{x-4}\), with \(x\ne-7,4\).
Mod 3 C
Multiplying and Dividing Rational Expressions
Multiply and divide rational expressions by factoring, canceling, and using reciprocals.
Quick reference
- Factor every numerator and denominator before simplifying.
- To divide rational expressions, multiply by the reciprocal of the divisor.
- Cancel common factors across the complete product.
- Record restrictions from every original denominator and from values that make a divisor zero.
- Write final exponents as positive values.
Example 1. Multiply \(\dfrac{5x^2}{y}\cdot\dfrac{2y}{9x}\).
Answer: \(\dfrac{10x}{9}\), with \(x\ne0\) and \(y\ne0\).
Example 2. Multiply \(\dfrac{30x^5}{x^9}\cdot\dfrac{6x^5}{5}\).
Answer: \(36x\), with \(x\ne0\).
Example 3. Multiply \(\dfrac{x}{7x-14}\cdot\dfrac{x^2-2x}{6}\).
Answer: \(\dfrac{x^2}{42}\), with \(x\ne2\).
Example 4. Multiply \(\dfrac{y^2+8y+15}{y^2+6y-40}\cdot\dfrac{y^2+2y-24}{y^2+6y+9}\).
Answer: \(\dfrac{(y+5)(y+6)}{(y+10)(y+3)}\), with \(y\ne-10,-3,4\).
Example 5. Divide \(\dfrac{7z^7}{3z^9}\div\dfrac{35z}{6z^4}\).
Answer: \(\dfrac{2z}{5}\), with \(z\ne0\).
Example 6. Divide \(\dfrac{x^2-9}{x^2-4x+3}\div\dfrac{x+3}{x-1}\).
Answer: \(1\), with \(x\ne-3,1,3\).
Mod 3 D
Adding and Subtracting Rational Expressions With Like Denominators
Combine rational expressions that already have a common denominator.
Quick reference
- When denominators match, add or subtract only the numerators.
- Use parentheses around an entire numerator when subtracting.
- Keep the common denominator, then factor and simplify if possible.
- Do not add denominators.
- Retain restrictions from the original denominator.
Example 1. Add \(\dfrac{6m}{5n}+\dfrac{4m}{5n}\).
Answer: \(\dfrac{2m}{n}\), with \(n\ne0\).
Example 2. Subtract \(\dfrac{4x}{2x-7}-\dfrac{14}{2x-7}\).
Answer: \(2\), with \(x\ne\dfrac72\).
Example 3. Add \(\dfrac{2}{y+4}+\dfrac{y+1}{y+4}\).
Answer: \(\dfrac{y+3}{y+4}\), with \(y\ne-4\).
Example 4. Subtract \(\dfrac{7x^2+6x}{x-4}-\dfrac{23x+44}{x-4}\).
Answer: \(7x+11\), with \(x\ne4\).
Example 5. Add \(\dfrac{6z+1}{z-1}+\dfrac{5z+2}{z-1}\).
Answer: \(\dfrac{11z+3}{z-1}\), with \(z\ne1\).
Mod 3 E
Adding and Subtracting Rational Expressions With Unlike Denominators
Find least common denominators and combine rational expressions with different denominators.
Quick reference
- Factor every denominator before choosing the least common denominator.
- The LCD contains each distinct factor raised to its greatest needed power.
- Multiply each fraction by the missing factors, then combine numerators.
- Distribute subtraction signs before combining like terms.
- Factor the resulting numerator and simplify only after the fractions are combined.
Example 1. Subtract \(\dfrac{8a}{b}-\dfrac{b}{4}\).
Answer: \(\dfrac{32a-b^2}{4b}\), with \(b\ne0\).
Example 2. Add \(\dfrac{6}{x}+\dfrac{7}{9x^2}\).
Answer: \(\dfrac{54x+7}{9x^2}\), with \(x\ne0\).
Example 3. Subtract \(\dfrac{3}{x+2}-\dfrac{2x}{x^2-4}\).
Answer: \(\dfrac{x-6}{(x-2)(x+2)}\), with \(x\ne-2,2\).
Example 4. Add \(\dfrac{3}{5x}+\dfrac{6}{x+7}\).
Answer: \(\dfrac{33x+21}{5x(x+7)}\), with \(x\ne0,-7\).
Example 5. Subtract \(\dfrac{x}{x^2-4}-\dfrac{5}{x^2-4x+4}\).
Answer: \(\dfrac{x(x-2)-5(x+2)}{(x-2)^2(x+2)}=\dfrac{x^2-7x-10}{(x-2)^2(x+2)}\), with \(x\ne-2,2\).
Example 6. Add \(\dfrac{x+8}{x^2+3x-28}+\dfrac{x+9}{x^2+5x-14}\).
Answer: \(\dfrac{2x^2+11x-52}{(x+7)(x-4)(x-2)}\), with \(x\ne-7,2,4\).
Mod 3 F
Rational Equations
Solve equations containing rational expressions and check restrictions and extraneous candidates.
Quick reference
- List excluded values before clearing denominators.
- Multiply every term by the least common denominator to remove fractions.
- Solve the resulting equation, which may be linear or quadratic.
- Reject any candidate excluded from the original equation.
- An equation may have one solution, multiple solutions, or no solution.
Example 1. Solve \(\dfrac{x}{4}+\dfrac{3x}{12}=\dfrac{x}{24}+5\).
Answer: \(x=12\).
Example 2. Solve \(4-\dfrac{8}{x}=16\).
Answer: \(x=-\dfrac23\).
Example 3. Solve \(6+\dfrac{x+9}{x}=10\).
Answer: \(x=3\).
Example 4. Solve \(\dfrac{15}{3y-8}=-1\).
Answer: \(y=-\dfrac73\).
Example 5. Solve \(\dfrac{7}{y-10}=\dfrac{9y}{2y(y-10)}\).
Answer: No solution. The algebra gives \(y=0\), but \(0\) is excluded.
Example 6. Solve \(\dfrac{2y}{2y+6}+\dfrac{4y-18}{3y+9}=\dfrac{8y+11}{y+3}\).
Answer: No solution. Clearing denominators gives \(y=-3\), but \(-3\) is excluded.
Mod 3 G
Solving Rational Equations With More Than One Variable
Rearrange rational formulas for a specified variable.
Quick reference
- Treat all variables except the requested one as known constants.
- Clear fractional forms by multiplying through by common denominators.
- Collect every term containing the requested variable on one side.
- Factor out the requested variable when it appears in multiple terms.
- State restrictions needed for any denominator in the final formula.
Example 1. Solve \(R=\dfrac{B}{L}\) for \(L\).
Answer: \(L=\dfrac{B}{R}\), with \(R\ne0\).
Example 2. Solve \(T=\dfrac{3R}{S}+\dfrac{U}{S}\) for \(S\).
Answer: \(S=\dfrac{3R+U}{T}\), with \(T\ne0\).
Example 3. Solve \(y=\dfrac{75C}{d^2}\) for \(C\).
Answer: \(C=\dfrac{yd^2}{75}\).
Example 4. Solve \(D=\dfrac{Q+K}{F}\) for \(F\).
Answer: \(F=\dfrac{Q+K}{D}\), with \(D\ne0\).
Example 5. Solve \(C=\dfrac{\pi t^2}{9}\) for \(t\), assuming \(t\ge0\).
Answer: \(t=3\sqrt{\dfrac{C}{\pi}}\).
Example 6. Solve \(\dfrac1s+\dfrac16=\dfrac1c\) for \(s\).
Answer: \(s=\dfrac{6c}{6-c}\), with \(c\ne0,6\).
Mod 3 H
Rational Equation Word Problems
Model reciprocal, work, motion, and proportion applications with rational equations.
Quick reference
- For work problems, a worker completing one job in \(t\) hours has rate \(1/t\) job per hour.
- Combined work rates add when people or machines work together.
- Motion applications use \(t=d/r\) when equal travel times are compared.
- Reciprocal relationships often translate directly into rational equations.
- Check units, restrictions, and whether the final value makes sense in context.
Example 1. Six times the reciprocal of a number equals three times the reciprocal of \(6\). Find the number.
Answer: \(\dfrac6x=\dfrac36\), so \(x=12\).
Example 2. One worker can pour a slab in \(3\) hours and another in \(2\) hours. How long do they take together?
Answer: \(\dfrac1t=\dfrac13+\dfrac12\), so \(t=\dfrac65\) hour, or \(1.2\) hours.
Example 3. A pilot flies \(468\) miles with the wind in the same time as \(360\) miles against it. The still-air speed is \(230\) mph. Find the wind speed.
Answer: \(\dfrac{468}{230+w}=\dfrac{360}{230-w}\), so \(w=30\) mph.
Example 4. A mixture uses \(12\) teaspoons of concentrate for \(3\) gallons of water. How much water is needed for \(16\) teaspoons?
Answer: \(\dfrac{12}{3}=\dfrac{16}{g}\), so \(g=4\) gallons.
Example 5. A custodian completes a job in \(8\) hours. With a second worker, the job takes \(4\) hours. How long would the second worker take alone?
Answer: \(\dfrac18+\dfrac1x=\dfrac14\), so \(x=8\) hours.
Example 6. Two pumps fill a tank together in \(6\) hours. One pump alone takes \(10\) hours. How long does the other take alone?
Answer: \(\dfrac1{10}+\dfrac1x=\dfrac16\), so \(x=15\) hours.
Mod 3 I
Square Roots and Cube Roots
Evaluate and simplify square and cube roots, including roots with variables.
Quick reference
- The principal square root \(\sqrt a\) is nonnegative.
- A negative number has no real square root, but negative numbers do have real cube roots.
- Recognize perfect square, cube, and fourth powers before taking a root.
- Apply the root to both the numerical coefficient and variable powers.
- Use the stated nonnegative or positive variable assumptions when simplifying variable roots.
Example 1. Find \(\sqrt{225}\).
Answer: \(15\).
Example 2. Simplify \(\sqrt{49x^6}\), assuming \(x\ge0\).
Answer: \(7x^3\).
Example 3. Find \(\sqrt[3]{64}\).
Answer: \(4\).
Example 4. Find \(\sqrt[3]{-216}\).
Answer: \(-6\).
Example 5. Simplify \(\sqrt[4]{81x^{16}}\), assuming \(x\ge0\).
Answer: \(3x^4\).
Example 6. Simplify \(\sqrt[3]{-27x^{12}y^6}\).
Answer: \(-3x^4y^2\).
Example 7. Classify \(\sqrt[4]{-81}\) in the real number system.
Answer: It is not a real number.
Mod 3 J
Evaluating Radical Functions
Evaluate square-root and cube-root functions at allowed inputs.
Quick reference
- Substitute the input into the entire radicand before evaluating the root.
- An even-index radical requires a nonnegative radicand for real-valued functions.
- Odd-index radicals accept every real radicand.
- Keep the radical index attached to the function when substituting.
- Simplify perfect powers and leave other real outputs in exact radical form.
Example 1. If \(f(x)=\sqrt{3x+7}\), find \(f(0)\).
Answer: \(f(0)=\sqrt7\).
Example 2. If \(g(x)=\sqrt[3]{x-13}\), find \(g(5)\).
Answer: \(g(5)=-2\).
Example 3. If \(g(x)=\sqrt[3]{x-11}\), find \(g(-16)\).
Answer: \(g(-16)=-3\).
Example 4. If \(f(x)=\sqrt{5x+2}\), find \(f(3)\).
Answer: \(f(3)=\sqrt{17}\).
Example 5. For \(h(t)=\sqrt[3]{2t+9}\), find \(h(-4)\).
Answer: \(h(-4)=1\).
Mod 3 K
Graphing Radical Functions
Find domains and graph square-root and cube-root functions from equations and tables.
Quick reference
- For a square-root function, require the radicand to be nonnegative to find the domain.
- The parent square-root function begins at \((0,0)\) and extends right.
- In \(\sqrt{x-h}+k\), the endpoint is \((h,k)\).
- Cube-root functions extend in both directions and have domain all real numbers.
- Use convenient inputs that make the radicand a perfect square or perfect cube when completing a table.
Example 1. Find the domain and graph \(f(x)=\sqrt{x}-5\).
Answer: Domain: \([0,\infty)\). Endpoint: \((0,-5)\).
A square-root curve starts at zero comma negative five and rises to the right
Example 2. Complete a table for \(f(x)=\sqrt{x-10}\) at \(x=10,11,14,19\), then graph it.
Answer: Outputs: \(0,1,2,3\). Domain: \([10,\infty)\).
| \(x\) | \(10\) | \(11\) | \(14\) | \(19\) |
|---|---|---|---|---|
| \(f(x)\) | \(0\) | \(1\) | \(2\) | \(3\) |
A square-root curve begins at ten comma zero and passes through the table points
Example 3. Find the domain and graph \(g(x)=\sqrt[3]{x}-3\).
Answer: Domain: \(( -\infty,\infty)\). The graph passes through \((0,-3)\).
A cube-root curve passes through zero comma negative three and extends in both directions
Example 4. Complete a table for \(h(x)=\sqrt[3]{x-8}\) at \(x=8,16,7,9,-19\), then graph it.
Answer: Outputs: \(0,2,-1,1,-3\). Domain: \(( -\infty,\infty)\).
| \(x\) | \(8\) | \(16\) | \(7\) | \(9\) | \(-19\) |
|---|---|---|---|---|---|
| \(h(x)\) | \(0\) | \(2\) | \(-1\) | \(1\) | \(-3\) |
A cube-root curve passes through the five table points and crosses the x-axis at eight
Mod 3 L
Rational Exponents
Write rational exponents in radical notation and simplify products and quotients with rational exponents.
Quick reference
- \(a^{1/n}=\sqrt[n]{a}\) and \(a^{m/n}=\sqrt[n]{a^m}\).
- The denominator of a rational exponent is the root index; the numerator is the power.
- Apply the product rule by adding rational exponents with a common denominator.
- Apply the quotient rule by subtracting rational exponents.
- Write the simplified result with a positive exponent.
Example 1. Write \((3x)^{2/3}\) in radical notation.
Answer: \(\sqrt[3]{(3x)^2}\).
Example 2. Write \((5x+4)^{3/4}\) in radical notation.
Answer: \(\sqrt[4]{(5x+4)^3}\).
Example 3. Simplify \(c^{3/5}c^{8/5}\).
Answer: \(c^{11/5}\).
Example 4. Simplify \(x^{-6/5}x^{11/5}\), assuming \(x>0\).
Answer: \(x\).
Example 5. Simplify \(\dfrac{y^{1/5}}{y^{1/10}}\), assuming \(y>0\).
Answer: \(y^{1/10}\).
Example 6. Simplify \(\dfrac{a^{7/6}}{a^{1/3}}\), assuming \(a>0\).
Answer: \(a^{5/6}\).
Mod 3 M
Product and Quotient Rules for Radicals
Multiply, divide, and simplify radical expressions using product and quotient properties.
Quick reference
- For appropriate real values, \(\sqrt a\sqrt b=\sqrt{ab}\).
- The quotient rule gives \(\sqrt{a/b}=\sqrt a/\sqrt b\) when defined.
- Multiply first, then extract perfect-power factors.
- The radical indices must match to use product or quotient rules directly.
- State variable assumptions or use absolute value when needed.
Example 1. Multiply \(\sqrt3\cdot\sqrt2\).
Answer: \(\sqrt6\).
Example 2. Multiply \(\sqrt{125}\cdot\sqrt{45}\).
Answer: \(75\).
Example 3. Multiply \(\sqrt[3]{3}\cdot\sqrt[3]{4}\).
Answer: \(\sqrt[3]{12}\).
Example 4. Multiply \(\sqrt{7x}\cdot\sqrt{3y}\), assuming positive variables.
Answer: \(\sqrt{21xy}\).
Example 5. Simplify \(\sqrt{\dfrac{2}{81}}\).
Answer: \(\dfrac{\sqrt2}{9}\).
Example 6. Simplify \(\sqrt{\dfrac{72x^5}{2x}}\), assuming \(x\ge0\).
Answer: \(6x^2\).
Mod 3 N
Distance Formula
Find exact and approximate distances between points in the coordinate plane.
Quick reference
- The distance formula is \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).
- Subtract coordinates in the same order; squaring removes sign differences.
- Simplify the radical for an exact answer before finding a decimal approximation.
- Horizontal distance is the absolute difference of x-coordinates; vertical distance uses y-coordinates.
- Distance is always nonnegative and should include units when given.
Example 1. Find the exact distance between \((-5,3)\) and \((2,-3)\), then approximate to the nearest thousandth.
Answer: \(\sqrt{85}\approx9.220\) units.
Example 2. Find the distance between \((-12,9)\) and \((-7,2)\).
Answer: \(\sqrt{74}\approx8.602\) units.
Example 3. Find the distance between \((0,\dfrac32)\) and \((-3,\dfrac25)\).
Answer: \(\dfrac{\sqrt{1021}}{10}\approx3.195\) units.
Example 4. Find the distance between \((4,-2)\) and \((4,11)\).
Answer: \(13\) units.
Example 5. A map uses coordinates \((2,5)\) and \((10,11)\) for two locations, with one coordinate unit equal to \(3\) miles. Find the actual distance.
Answer: Coordinate distance: \(10\) units. Actual distance: \(30\) miles.
Mod 3 O
Midpoint Formula
Find the point halfway between two endpoints in the coordinate plane.
Quick reference
- The midpoint formula is \(M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\).
- Average the x-coordinates and y-coordinates separately.
- Keep fractional coordinates in simplified exact form.
- Average decimal coordinates in the same way as integer coordinates.
- The midpoint lies on the segment and is the same distance from both endpoints.
Example 1. Find the midpoint of \((4,-9)\) and \((6,3)\).
Answer: \((5,-3)\).
Example 2. Find the midpoint of \((-2,-2)\) and \((-10,7)\).
Answer: \((-6,\dfrac52)\).
Example 3. Find the midpoint of \((7,3)\) and \((-2,-3)\).
Answer: \((\dfrac52,0)\).
Example 4. Find the midpoint of \((-\dfrac56,\dfrac67)\) and \((-\dfrac76,\dfrac27)\).
Answer: \((-1,\dfrac47)\).
Example 5. Find the midpoint of \((-5.5,4.2)\) and \((-9.2,0.7)\).
Answer: \((-7.35,2.45)\).
Mod 3 P
Adding and Subtracting Radical Expressions
Simplify radicals and combine like radical terms.
Quick reference
- Like radicals have the same index and the same simplified radicand.
- Simplify every radical before deciding which terms are like.
- Add or subtract coefficients while keeping the common radical part.
- Radicals with different simplified radicands cannot be combined.
- Apply the same process to variable radicals under the stated assumptions.
Example 1. Simplify \(\sqrt{180}-\sqrt{20}\).
Answer: \(4\sqrt5\).
Example 2. Simplify \(9\sqrt{3x^3}+3x\sqrt{27x}\), assuming \(x>0\).
Answer: \(18x\sqrt{3x}\).
Example 3. Simplify \(5\sqrt{75}-4\sqrt{50}+\sqrt{12}\).
Answer: \(27\sqrt3-20\sqrt2\).
Example 4. Simplify \(\sqrt{36b^3}+\sqrt{16b^3}-\sqrt{25b^3}\), assuming \(b>0\).
Answer: \(5b\sqrt b\).
Example 5. Simplify \(-4\sqrt{625}+3\sqrt{40}\).
Answer: \(-100+6\sqrt{10}\).
Example 6. A trapezoid has side lengths \(7\sqrt2\), \(2\sqrt{18}\), \(\sqrt{18}\), and \(2\sqrt{50}\) inches. Find its perimeter.
Answer: \(7\sqrt2+6\sqrt2+3\sqrt2+10\sqrt2=26\sqrt2\) inches.
Mod 3 Q
Multiplying Radical Expressions
Multiply monomial and binomial radical expressions and simplify the results.
Quick reference
- Use distribution or FOIL when multiplying sums containing radicals.
- Multiply coefficients and radicands, then simplify.
- \((\sqrt a)^2=a\) when the principal root is defined.
- Conjugates \((a+b)(a-b)\) create a difference of squares.
- Combine like terms after all products are simplified.
Example 1. Multiply \(7(\sqrt{10}+\sqrt3)\).
Answer: \(7\sqrt{10}+7\sqrt3\).
Example 2. Multiply \((-7\sqrt2)(2\sqrt2)\).
Answer: \(-28\).
Example 3. Multiply \((2-\sqrt{3x})(9+2\sqrt{3x})\), assuming \(x\ge0\).
Answer: \(18-5\sqrt{3x}-6x\).
Example 4. Multiply \((9-\sqrt{55})(9+\sqrt{55})\).
Answer: \(26\).
Example 5. Multiply \((\sqrt5+3)^2\).
Answer: \(14+6\sqrt5\).
Example 6. Multiply \((2\sqrt x-\sqrt y)(3\sqrt x+4\sqrt y)\).
Answer: \(6x+5\sqrt{xy}-4y\).
Mod 3 R
Rationalizing Denominators
Rewrite radical fractions so no radical remains in the denominator.
Quick reference
- For a binomial denominator containing a radical, multiply by its conjugate.
- Use the opposite middle sign when writing the conjugate.
- Conjugates have the same terms with the middle sign changed.
- The denominator becomes a difference of squares.
- Simplify the numerator and denominator completely after rationalizing.
Example 1. Rationalize \(\dfrac{9}{2-\sqrt7}\).
Answer: \(-6-3\sqrt7\).
Example 2. Rationalize \(\dfrac{-8}{\sqrt x-3}\).
Answer: \(\dfrac{-8(\sqrt x+3)}{x-9}\), with \(x\ge0\) and \(x\ne9\).
Example 3. Rationalize \(\dfrac{2-\sqrt3}{2+\sqrt3}\).
Answer: \(7-4\sqrt3\).
Example 4. Rationalize \(\dfrac{5}{1+\sqrt3}\).
Answer: \(\dfrac{5(\sqrt3-1)}{2}\).
Example 5. Rationalize \(\dfrac{6}{\sqrt5+2}\).
Answer: \(6\sqrt5-12\).
Mod 3 S
Solving Radical Equations
Isolate radicals, remove them with powers, and reject extraneous solutions.
Quick reference
- Isolate one radical before raising both sides to a power.
- Square both sides to remove a square root; cube both sides to remove a cube root.
- Simplify and isolate again when an equation contains two radicals.
- Even powers can introduce extraneous solutions.
- Check every candidate in the original radical equation.
Example 1. Solve \(\sqrt{2x}=4\).
Answer: \(x=8\).
Example 2. Solve \(\sqrt{x-3}=6\).
Answer: \(x=39\).
Example 3. Solve \(\sqrt{5x-9}-4=0\).
Answer: \(x=5\).
Example 4. Solve \(\sqrt{x+5}=x-1\).
Answer: \(x=4\); the other algebraic candidate is extraneous.
Example 5. Solve \(\sqrt{x+1}+1=\sqrt{2x+4}\).
Answer: \(x=0\).
Example 6. Solve \(\sqrt[3]{2x-1}=3\).
Answer: \(x=14\).
Example 7. Solve \(\sqrt{x-2}=-3\).
Answer: No solution, \(\varnothing\).
Mod 3 T
Complex Numbers
Write and operate with complex numbers using the imaginary unit i.
Quick reference
- The imaginary unit is defined by \(i^2=-1\), so \(\sqrt{-a}=i\sqrt a\) for \(a>0\).
- Combine real parts with real parts and imaginary parts with imaginary parts.
- Multiply using distribution and replace \(i^2\) with \(-1\).
- To divide complex numbers, multiply by the conjugate of the denominator.
- Powers of \(i\) repeat every four exponents.
Example 1. Express \(\sqrt{-121}\) in terms of \(i\).
Answer: \(11i\).
Example 2. Simplify \(\sqrt{-63}\).
Answer: \(3i\sqrt7\).
Example 3. Add \((7-8i)+(3+4i)\).
Answer: \(10-4i\).
Example 4. Subtract \((2+6i)-(3-4i)\).
Answer: \(-1+10i\).
Example 5. Multiply \((-7i)(-10i)\).
Answer: \(-70\).
Example 6. Multiply \(2i(5-2i)\).
Answer: \(4+10i\).
Example 7. Write \(\dfrac{4}{2+8i}\) in \(a+bi\) form.
Answer: \(\dfrac2{17}-\dfrac8{17}i\).
Example 8. Find \((-2i)^9\).
Answer: \(-512i\).
Module 4: Quadratic Methods, Inequalities, Graphs, and Applications
Finish quadratic equation methods, solve polynomial and rational inequalities, graph quadratics in multiple forms, and interpret quadratic models.
Mod 4 A
Solving Quadratic Equations Using the Square Root Method
Use the square-root property to solve equations with an isolated squared expression.
Quick reference
- If \(u^2=k\) with \(k>0\), then \(u=\pm\sqrt k\).
- Isolate the squared expression before taking square roots.
- Remember the \(\pm\) when solving an equation; the principal radical symbol alone is nonnegative.
- Simplify exact radical answers before listing both real solutions.
- Check both candidates in the original equation.
Example 1. Solve \(x^2=9\).
Answer: \(x=-3,3\).
Example 2. Solve \(x^2-11=0\).
Answer: \(x=\pm\sqrt{11}\).
Example 3. Solve \(x^2=98\).
Answer: \(x=\pm7\sqrt2\).
Example 4. Solve \((x-4)^2=25\).
Answer: \(x=-1,9\).
Example 5. Solve \(3(x+2)^2=42\).
Answer: \(x=-2\pm\sqrt{14}\).
Example 6. Solve \((3x+1)^2=8\).
Answer: \(x=\dfrac{-1\pm2\sqrt2}{3}\).
Mod 4 B
Solving Quadratic Equations by Completing the Square
Create perfect-square trinomials and solve quadratic equations by completing the square.
Quick reference
- For \(x^2+bx\), add \((b/2)^2\) to create \((x+b/2)^2\).
- Add the same value to both sides of the equation.
- Rewrite the left side as a perfect-square binomial after adding \((b/2)^2\).
- Use the square-root property after rewriting one side as a perfect square.
- Keep real radical answers in exact simplified form.
Example 1. Solve \(x^2+12x=-27\) by completing the square.
Answer: \(x=-3,-9\).
Example 2. Solve \(x^2+8x=-7\) by completing the square.
Answer: \(x=-1,-7\).
Example 3. Solve \(x^2+8x+11=0\) by completing the square.
Answer: \(x=-4\pm\sqrt5\).
Example 4. Solve \(x^2-6x+1=0\) by completing the square.
Answer: \(x=3\pm2\sqrt2\).
Example 5. Solve \(x^2+2x-5=0\) by completing the square.
Answer: \(x=-1\pm\sqrt6\).
Example 6. Solve \(x^2+6x+6=0\) by completing the square.
Answer: \(x=-3\pm\sqrt3\).
Mod 4 C
Solving Quadratic Equations Using the Quadratic Formula
Use the quadratic formula to solve quadratic equations.
Quick reference
- For \(ax^2+bx+c=0\), \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\).
- Write the equation in standard form and identify \(a\), \(b\), and \(c\) with their signs.
- Evaluate \(b^2-4ac\) carefully before simplifying the square root.
- Simplify the radical and reduce the full fraction when possible.
- Use exact answers unless a decimal approximation is requested.
Example 1. Solve \(m^2-4m-12=0\) using the quadratic formula.
Answer: \(m=-2,6\).
Example 2. Solve \(5y=3y^2-8\).
Answer: \(y=-1,\dfrac83\).
Example 3. Solve \(x^2-8x+16=0\).
Answer: \(x=4\), a repeated solution.
Example 4. Solve \(2x^2+3x-7=0\).
Answer: \(x=\dfrac{-3\pm\sqrt{65}}{4}\).
Example 5. Solve \(x^2+2x+10=0\).
Answer: \(x=-1\pm3i\).
Example 6. Solve \(9m^2-2m=10\) using the quadratic formula.
Answer: \(m=\dfrac{1\pm\sqrt{91}}{9}\).
Mod 4 D
Solving Equations in Quadratic Form
Solve radical, rational, and quartic equations that lead to quadratic equations.
Quick reference
- For a radical equation, isolate the radical and square both sides before solving the resulting quadratic.
- Check every radical-equation candidate because squaring can introduce an extraneous solution.
- For a rational equation, list excluded values and multiply through by the least common denominator.
- For a quartic in \(x^2\), let \(u=x^2\), solve the quadratic in \(u\), and then solve for \(x\).
- Check restrictions and every candidate in the original equation.
Example 1. Solve \(3x=\sqrt{1+8x}\).
Answer: Squaring gives \(9x^2-8x-1=0\). The candidates are \(1\) and \(-\dfrac19\), but only \(x=1\) checks.
Example 2. Solve \(\sqrt{100x}=x+16\).
Answer: Squaring gives \(x^2-68x+256=0\), so \(x=4,64\). Both solutions check.
Example 3. Solve \(\dfrac2x+\dfrac3{x-3}=1\).
Answer: The restrictions are \(x\ne0,3\). Clearing denominators gives \(x^2-8x+6=0\), so \(x=4\pm\sqrt{10}\).
Example 4. Solve \(\dfrac4{x^2-3x+2}=\dfrac{3x}{x-1}-\dfrac{x}{x-2}\).
Answer: The restrictions are \(x\ne1,2\). Clearing denominators gives solutions \(x=\dfrac{5\pm\sqrt{57}}4\).
Example 5. Solve \(p^4-16=0\).
Answer: \(p=-2,2,-2i,2i\).
Example 6. Solve \(z^4-20z^2+64=0\).
Answer: Let \(u=z^2\). Then \((u-4)(u-16)=0\), so \(z=-4,-2,2,4\).
Mod 4 E
Polynomial and Rational Inequalities
Use critical values and sign analysis to solve polynomial and rational inequalities.
Quick reference
- Move all terms to one side and factor when possible.
- Critical values are zeros of the numerator and, for rational expressions, excluded zeros of the denominator.
- Use critical values to divide the number line into test intervals.
- Include zeros for \(\le\) or \(\ge\), but never include denominator zeros.
- Express the selected sign intervals with interval notation.
Example 1. Solve \((x+4)(x+2)>0\).
Answer: \(( -\infty,-4)\cup(-2,\infty)\).
Open points at negative four and negative two with shading outside
Example 2. Solve \((x-9)(x+1)\le0\).
Answer: \([-1,9]\).
Closed points at negative one and nine with shading between
Example 3. Solve \(x^2+2x-3\le0\).
Answer: \([-3,1]\).
Example 4. Solve \(x(x-2)(x+4)\le0\).
Answer: \(( -\infty,-4]\cup[0,2]\).
Example 5. Solve \(\dfrac{x-3}{x+2}>0\).
Answer: \(( -\infty,-2)\cup(3,\infty)\).
Example 6. Solve \(\dfrac{(x+1)(x-4)}{x-2}\le0\).
Answer: \(( -\infty,-1]\cup(2,4]\).
Example 7. Solve \(\dfrac{x+5}{(x-1)^2}\ge0\).
Answer: \([-5,1)\cup(1,\infty)\).
Mod 4 F
Graphing Quadratic Functions in Vertex Form
Graph quadratics in vertex form and identify the vertex and axis of symmetry.
Quick reference
- Vertex form is \(f(x)=a(x-h)^2+k\), with vertex \((h,k)\).
- The axis of symmetry is \(x=h\).
- If \(a>0\), the parabola opens up; if \(a<0\), it opens down.
- Plot the vertex first and use symmetric points on both sides of the axis.
- Draw the parabola as a solid curve and the axis of symmetry as a dashed vertical line.
Example 1. Graph \(f(x)=x^2-2\). State the vertex and axis of symmetry.
Answer: Vertex: \((0,-2)\). Axis: \(x=0\). Opens upward.
An upward-opening parabola has vertex at zero comma negative two
Example 2. Graph \(h(x)=(x+1)^2\). State the vertex and axis.
Answer: Vertex: \((-1,0)\). Axis: \(x=-1\).
An upward-opening parabola has vertex at negative one comma zero
Example 3. Graph \(g(x)=(x-6)^2\). State the vertex and axis of symmetry.
Answer: Vertex: \((6,0)\). Axis: \(x=6\).
An upward-opening parabola has vertex at six comma zero
Example 4. Graph \(p(x)=(x-2)^2+3\). State the vertex and axis of symmetry.
Answer: Vertex: \((2,3)\). Axis: \(x=2\).
An upward-opening parabola has vertex at two comma three
Example 5. Graph \(q(x)=-2(x+6)^2+4\). State the vertex and axis of symmetry.
Answer: Vertex: \((-6,4)\). Axis: \(x=-6\).
A downward-opening parabola has vertex at negative six comma four
Mod 4 G
Graphing Quadratic Functions Continued
Analyze and graph quadratic functions in standard form using vertices, intercepts, and symmetry.
Quick reference
- For \(f(x)=ax^2+bx+c\), the axis of symmetry is \(x=-b/(2a)\).
- Substitute the axis value to find the vertex.
- The y-intercept is \((0,c)\); solve \(ax^2+bx+c=0\) for x-intercepts.
- Use symmetry to plot matching points on opposite sides of the axis.
- Use the sign of \(a\) to determine whether the graph opens upward or downward.
Example 1. Analyze and graph \(f(x)=x^2+6x+5\).
Answer: Vertex: \((-3,-4)\). Axis: \(x=-3\). Opens up. x-intercepts: \((-5,0)\), \((-1,0)\). y-intercept: \((0,5)\).
An upward-opening parabola has vertex at negative three comma negative four and zeros at negative five and negative one
Example 2. Analyze and graph \(g(x)=-x^2+4x+5\).
Answer: Vertex: \((2,9)\). Axis: \(x=2\). Opens down. x-intercepts: \((-1,0)\), \((5,0)\). y-intercept: \((0,5)\).
A downward-opening parabola has vertex at two comma nine and zeros at negative one and five
Example 3. Analyze and graph \(h(x)=2x^2-8x+8\).
Answer: Vertex: \((2,0)\). Axis: \(x=2\). Opens up. Repeated x-intercept: \((2,0)\). y-intercept: \((0,8)\).
An upward-opening parabola touches the x-axis at its vertex two comma zero
Example 4. Analyze \(q(x)=x^2+2x+5\). Does it have real x-intercepts?
Answer: Vertex: \((-1,4)\). It opens up and has no real x-intercepts because its minimum value is \(4\).
An upward-opening parabola has vertex above the x-axis at negative one comma four
Example 5. Analyze and graph \(r(x)=3x^2+12x+13\).
Answer: Vertex: \((-2,1)\). Axis: \(x=-2\). Opens up. There are no real x-intercepts. y-intercept: \((0,13)\).
An upward-opening parabola has vertex at negative two comma one and stays above the x-axis
Mod 4 H
Applications of Quadratic Functions
Use quadratic vertices to solve maximum and minimum application problems.
Quick reference
- A quadratic vertex gives a maximum when the parabola opens down and a minimum when it opens up.
- Use \(x=-b/(2a)\) to find the input at the vertex, then evaluate the model.
- Projectile models use the vertex height to answer maximum-height questions.
- Cost, number-product, and rectangle-area models require units and a contextual interpretation.
- Restrict the domain to values that make sense and reject nonphysical answers.
Example 1. A projectile is launched from the ground at \(96\) ft/s, with height \(h(t)=-16t^2+96t\). Find its maximum height.
Answer: The maximum height is \(144\) feet.
Example 2. Manufacturing cost is \(C(x)=4x^2-1600x+170{,}500\). How many units minimize cost, and what is the minimum cost?
Answer: The minimum occurs at \(x=200\) units, with cost \(\$10{,}500\).
Example 3. Find two numbers whose sum is \(82\) and whose product is as large as possible.
Answer: The numbers are \(41\) and \(41\); their maximum product is \(1681\).
Example 4. Find two numbers whose difference is \(56\) and whose product is as small as possible.
Answer: The numbers are \(28\) and \(-28\); their minimum product is \(-784\).
Example 5. The length and width of a rectangle have a sum of \(44\). Find the dimensions that maximize its area.
Answer: The rectangle is \(22\) by \(22\), with maximum area \(484\) square units.
Example 6. Find two numbers whose sum is \(60\) and whose product is as large as possible.
Answer: The numbers are \(30\) and \(30\); their maximum product is \(900\).
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