Polynomial Equations

We have spent considerable time learning how to factor polynomials. We will now look at polynomial equations and solve them using factoring, if possible.

A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.

We have already solved polynomial equations of degree one. Polynomial equations of degree one are linear equations are of the form $a x + b = c .$

We are now going to solve polynomial equations of degree two. A polynomial equation of degree two is called a quadratic equation. Listed below are some examples of quadratic equations:

x^{2} + 5 x + 6 = 0 3 y^{2} + 4 y = 10 64 u^{2} - 81 = 0 n (n + 1) = 42

The last equation doesn’t appear to have the variable squared, but when we simplify the expression on the left we will get $n^{2} + n .$

The general form of a quadratic equation is $a x^{2} + b x + c = 0,$ with $a \neq 0 .$ (If $a = 0,$ then $0 \cdot x^{2} = 0$ and we are left with no quadratic term.)

To solve quadratic equations we need methods different from the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.

Use the Zero Product Property

We will first solve some quadratic equations by using the Zero Product Property. The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.

We will now use the Zero Product Property, to solve a quadratic equation.

How to Solve a Quadratic Equation Using the Zero Product Property

Solve: $(5 n - 2) (6 n - 1) = 0 .$

Solution

The equation is open parentheses 5n minus 2 close parentheses open parentheses 6n minus 1 close parentheses equals 0. The product equals zero, so at least one factor must equal zero. Step 1 is set each factor equal to zero. So, 5n minus 2 equals 0 and 6n minus 1 equals 0.

Step 2 is to solve the linear equations. So, we get n equal to 2 by 5 and n equal to 1 by 6.

Step 3 is to check by substituting each solution separately into the original equation.

Solve Quadratic Equations by Factoring

The Zero Product Property works very nicely to solve quadratic equations. The quadratic equation must be factored, with zero isolated on one side. So we must be sure to start with the quadratic equation in standard form, $a x^{2} + b x + c = 0 .$ Then we must factor the expression on the left.

How to Solve a Quadratic Equation by Factoring

Solve: $2 y^{2} = 13 y + 45 .$

Solution

The equation is 2 y squared equals 13y plus 45. Step 1 is to write it in standard form a x squared plus bx plus c. So we have 2 y squared minus 13y minus 45 equals 0.

Step 2 is to factor the quadratic expression. So we have 2y plus 5, y minus 9 equals 0.

Step 3 is to use the zero product property. Setting each factor equal to zero, we have two linear equations: 2y plus 5 equals 0 and y minus 9 equals 0.

Step 4 is to solve the linear equations. We get, y equals minus 5 by 2 and y equals 9.

Step 5 is to check by substituting each solution separately into the original equation

Before we factor, we must make sure the quadratic equation is in standard form.

Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?

Solve: $169 x^{2} = 49 .$

Solution

Step-by-step solution for the quadratic equation 169x^2 = 49 using factoring and the Zero Product Property.
	$169 x^{2} = 49$
Write the quadratic equation in standard form.	$169 x^{2} - 49 = 0$
Factor. It is a difference of squares.	$(13 x - 7) (13 x + 7) = 0$
Use the Zero Product Property to set each factor to $0$ . Solve each equation.	$\begin{matrix} 13 x - 7 & = & 0 & 13 x + 7 & = & 0 \\ 13 x & = & 7 & 13 x & = & −7 \\ x & = & \frac{7}{13} & x & = & - \frac{7}{13} \end{matrix}$

Check:

We leave the check up to you.

In the next example, the left side of the equation is factored, but the right side is not zero. In order to use the Zero Product Property, one side of the equation must be zero. We’ll multiply the factors and then write the equation in standard form.

Solve: $(3 x - 8) (x - 1) = 3 x .$

Solution

This table demonstrates the step-by-step process of solving a quadratic equation by factoring, showing each mathematical transformation.
	$(3 x - 8) (x - 1) = 3 x$
Multiply the binomials.	$3 x^{2} - 11 x + 8 = 3 x$
Write the quadratic equation in standard form.	$3 x^{2} - 14 x + 8 = 0$
Factor the trinomial.	$(3 x - 2) (x - 4) = 0$
Use the Zero Product Property to set each factor to 0. Solve each equation.	$\begin{matrix} 3 x - 2 & = & 0 & x - 4 & = & 0 \\ 3 x & = & 2 & x & = & 4 \end{matrix}$
	$x = \frac{2}{3}$
Check your answers.	The check is left to you.

In the next example, when we factor the quadratic equation we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.

Solve: $3 x^{2} = 12 x + 63 .$

Solution

Step-by-step method for solving a quadratic equation by factoring, with corresponding mathematical expressions.
	$3 x^{2} = 12 x + 63$
Write the quadratic equation in standard form.	$3 x^{2} - 12 x - 63 = 0$
Factor the greatest common factor first.	$3 (x^{2} - 4 x - 21) = 0$
Factor the trinomial.	$3 (x - 7) (x + 3) = 0$
Use the Zero Product Property to set each factor to 0. Solve each equation.	$\begin{matrix} 3 \neq 0 & x - 7 & = & 0 & x + 3 & = & 0 \\ 3 \neq 0 & x & = & 7 & x & = & −3 \end{matrix}$
Check your answers.	The check is left to you.

The Zero Product Property also applies to the product of three or more factors. If the product is zero, at least one of the factors must be zero. We can solve some equations of degree greater than two by using the Zero Product Property, just like we solved quadratic equations.

Solve: $9 m^{3} + 100 m = 60 m^{2} .$

Solution

Solution steps for the cubic equation 9m^3 + 100m = 60m^2, demonstrating factoring and the Zero Product Property.
	$9 m^{3} + 100 m = 60 m^{2}$
Bring all the terms to one side so that the other side is zero.	$9 m^{3} - 60 m^{2} + 100 m = 0$
Factor the greatest common factor first.	$m (9 m^{2} - 60 m + 100) = 0$
Factor the trinomial.	$m (3 m - 10) (3 m - 10) = 0$
Use the Zero Product Property to set each factor to 0. Solve each equation.	$\begin{matrix} m = 0 & 3 m - 10 & = & 0 & 3 m - 10 & = & 0 \\ m = 0 & m & = & \frac{10}{3} & m & = & \frac{10}{3} \end{matrix}$
Check your answers.	The check is left to you.

Solve Equations with Polynomial Functions

As our study of polynomial functions continues, it will often be important to know when the function will have a certain value or what points lie on the graph of the function. Our work with the Zero Product Property will be help us find these answers.

For the function $f (x) = x^{2} + 2 x - 2,$

ⓐ find x when $f (x) = 6$ ⓑ find two points that lie on the graph of the function.

Solution

ⓐ

Demonstrates solving a quadratic equation (x^2 + 2x - 2 = 6) by factoring, with steps and solution verification.
	$f (x) = x^{2} + 2 x - 2$
Substitute 6 for $f (x)$ .	$6 = x^{2} + 2 x - 2$
Put the quadratic in standard form.	$x^{2} + 2 x - 8 = 0$
Factor the trinomial.	$(x + 4) (x - 2) = 0$
Use the zero product property. Solve.	$\begin{matrix} x + 4 & = & 0 & or & x - 2 & = & 0 \\ x & = & −4 & or & x & = & 2 \end{matrix}$
Check:
$\begin{matrix} f (x) = x^{2} + 2 x - 2 & f (x) = x^{2} + 2 x - 2 \\ f (− 4) = {(− 4)}^{2} + 2 (− 4) - 2 & f (2) = 2^{2} + 2 \cdot 2 - 2 \\ f (− 4) = 16 - 8 - 2 & f (2) = 4 + 4 - 2 \\ f (− 4) = 6 ✓ & f (2) = 6 ✓ \end{matrix}$

ⓑ Since

f (−4) = 6

and

f (2) = 6,

the points

(−4, 6)

and

(2, 6)

lie on the graph of the function.

The Zero Product Property also helps us determine where the function is zero. A value of x where the function is 0, is called a zero of the function.

When $f (x) = 0,$ the point $(x, 0)$ is a point on the graph. This point is an x-intercept of the graph. It is often important to know where the graph of a function crosses the axes. We will see some examples later.

For the function $f (x) = 3 x^{2} + 10 x - 8,$ find

ⓐ the zeros of the function, ⓑ any x-intercepts of the graph of the function, ⓒ any y-intercepts of the graph of the function

Solution

ⓐ To find the zeros of the function, we need to find when the function value is 0.

Step-by-step solution to the quadratic equation 3x^2 + 10x - 8 = 0 by factoring, showing the process from substitution to finding the roots.
	$f (x) = 3 x^{2} + 10 x - 8$
Substitute 0 for $f (x)$ .	$0 = 3 x^{2} + 10 x - 8$
Factor the trinomial.	$(x + 4) (3 x - 2) = 0$
Use the zero product property. Solve.	$\begin{matrix} x + 4 & = & 0 & or & 3 x - 2 & = & 0 \\ x & = & −4 & or & x & = & \frac{2}{3} \end{matrix}$

ⓑ An x-intercept occurs when

y = 0 .

Since

f (−4) = 0

and

f (\frac{2}{3}) = 0,

the points

(−4, 0)

and

(\frac{2}{3}, 0)

lie on the graph. These points are x-intercepts of the function.

ⓒ A y-intercept occurs when

x = 0 .

To find the y-intercepts we need to find

f (0) .

Step-by-step evaluation of the function f(x) = 3x^2 + 10x - 8 for x=0, demonstrating substitution and simplification to find f(0).
	$f (x) = 3 x^{2} + 10 x - 8$
Find $f (0)$ by substituting 0 for $x$ .	$f (0) = 3 \cdot 0^{2} + 10 \cdot 0 - 8$
Simplify.	$f (0) = −8$

Since

f (0) = −8,

the point

(0, −8)

lies on the graph. This point is the y-intercept of the function.

Solve Applications Modeled by Polynomial Equations

The problem-solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to polynomial equations. We will copy the problem-solving strategy here so we can use it for reference.

We will start with a number problem to get practice translating words into a polynomial equation.

The product of two consecutive odd integers is 323. Find the integers.

Solution

This table outlines the step-by-step process for solving a word problem to find consecutive odd integers whose product is 323, including variable definition, equation setup, and solution verification.
Step 1. Read the problem.
Step 2. Identify what we are looking for.	We are looking for two consecutive integers.
Step 3. Name what we are looking for.	Let $n = the first integer.$
	$n + 2 = next consecutive odd integer$
Step 4. Translate into an equation. Restate the problem in a sentence.	The product of the two consecutive odd integers is 323.
	$n (n + 2) = 323$
Step 5. Solve the equation.	$n^{2} + 2 n = 323$
Bring all the terms to one side.	$n^{2} + 2 n - 323 = 0$
Factor the trinomial.	$(n - 17) (n + 19) = 0$
Use the Zero Product Property. Solve the equations.	$\begin{matrix} n - 17 & = & 0 & n + 19 & = & 0 \\ n & = & 17 & n & = & −19 \end{matrix}$
There are two values for n that are solutions to this problem. So there are two sets of consecutive odd integers that will work.
If the first integer is $n = 17$	If the first integer is $n = −19$
then the next odd integer is	then the next odd integer is
$n + 2$	$n + 2$
$17 + 2$	$- 19 + 2$
$19$	$- 17$
$17, 19$	$- 17, −19$
Step 6. Check the answer.
The results are consecutive odd integers
$17, 19 and − 19, −17 .$
$17 \cdot 19 = 323 ✓ − 19 (− 17) = 323 ✓$
Both pairs of consecutive integers are solutions.
Step 7. Answer the question	The consecutive integers are 17, 19 and $- 19, −17 .$

Were you surprised by the pair of negative integers that is one of the solutions to the previous example? The product of the two positive integers and the product of the two negative integers both give positive results.

In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.

A rectangular bedroom has an area 117 square feet. The length of the bedroom is four feet more than the width. Find the length and width of the bedroom.

Solution

Step 1. Read the problem. In problems involving geometric figures, a sketch can help you visualize the situation.
Step 2. Identify what you are looking for.	We are looking for the length and width.
Step 3. Name what you are looking for.	Let $w =$ the width of the bedroom.
The length is four feet more than the width.	$w + 4 =$ the length of the garden
Step 4. Translate into an equation.
Restate the important information in a sentence.	The area of the bedroom is 117 square feet.
Use the formula for the area of a rectangle.	$A = l \cdot w$
Substitute in the variables.	$117 = (w + 4) w$
Step 5. Solve the equation Distribute first.	$117 = w^{2} + 4 w$
Get zero on one side.	$117 = w^{2} + 4 w$
Factor the trinomial.	$0 = w^{2} + 4 w - 117$
Use the Zero Product Property.	$0 = (w + 13) (w - 9)$
Solve each equation.	$0 = w + 13 0 = w - 9$
Since w is the width of the bedroom, it does not make sense for it to be negative. We eliminate that value for w.	$−13 = w 9 = w$
	$w = 9$ Width is 9 feet.
Find the value of the length.	$w + 4$ $9 + 4$ 13 Length is 13 feet.
Step 6. Check the answer. Does the answer make sense? Yes, this makes sense.
Step 7. Answer the question.	The width of the bedroom is 9 feet and the length is 13 feet.

In the next example, we will use the Pythagorean Theorem $(a^{2} + b^{2} = c^{2}) .$ This formula gives the relation between the legs and the hypotenuse of a right triangle.

Figure shows a right triangle with the shortest side being a, the second side being b and the hypotenuse being c.

We will use this formula in the next example.

A boat’s sail is in the shape of a right triangle as shown. The hypotenuse will be 17 feet long. The length of one side will be 7 feet less than the length of the other side. Find the lengths of the sides of the sail.

Figure shows a right triangle with the shortest side being x, the second side being x minus 7 and the hypotenuse being 17.

Solution

Step 1. Read the problem
Step 2. Identify what you are looking for.	We are looking for the lengths of the sides of the sail.
Step 3. Name what you are looking for. One side is 7 less than the other.	Let $x =$ length of a side of the sail. $x - 7 =$ length of other side
Step 4. Translate into an equation. Since this is a right triangle we can use the Pythagorean Theorem.	$a^{2} + b^{2} = c^{2}$
Substitute in the variables.	$x^{2} + {(x - 7)}^{2} = 17^{2}$
Step 5. Solve the equation Simplify.	$x^{2} + x^{2} - 14 x + 49 = 289$
	$2 x^{2} - 14 x + 49 = 289$
It is a quadratic equation, so get zero on one side.	$2 x^{2} - 14 x - 240 = 0$
Factor the greatest common factor.	$2 (x^{2} - 7 x - 120) = 0$
Factor the trinomial.	$2 (x - 15) (x + 8) = 0$
Use the Zero Product Property.	$2 \neq 0 x - 15 = 0 x + 8 = 0$
Solve.	$2 \neq 0 x = 15 x = −8$
Since x is a side of the triangle, $x = −8$ does not make sense.	$2 \neq 0 x = 15 x = −8$
Find the length of the other side.
If the length of one side is then the length of the other side is	8 is the length of the other side.
Step 6. Check the answer in the problem Do these numbers make sense?
Step 7. Answer the question	The sides of the sail are 8, 15 and 17 feet.

The next example uses the function that gives the height of an object as a function of time when it is thrown from 80 feet above the ground.

Dennis is going to throw his rubber band ball upward from the top of a campus building. When he throws the rubber band ball from 80 feet above the ground, the function $h (t) = −16 t^{2} + 64 t + 80$ models the height, h, of the ball above the ground as a function of time, t. Find:

ⓐ the zeros of this function which tell us when the ball hits the ground, ⓑ when the ball will be 80 feet above the ground, ⓒ the height of the ball at $t = 2$ seconds.

Solution

ⓐ The zeros of this function are found by solving

h (t) = 0 .

This will tell us when the ball will hit the ground.

Step-by-step solution of the quadratic equation -16t^2 + 64t + 80 = 0 by factoring.
	$h (t) = 0$
Substitute in the polynomial for $h (t) .$	$−16 t^{2} + 64 t + 80 = 0$
Factor the GCF, $−16 .$	$−16 (t^{2} - 4 t - 5) = 0$
Factor the trinomial.	$−16 (t - 5) (t + 1) = 0$
Use the Zero Product Property. Solve.	$\begin{matrix} t - 5 & = & 0 & t + 1 & = & 0 \\ t & = & 5 & t & = & −1 \end{matrix}$

The result $t = 5$ tells us the ball will hit the ground 5 seconds after it is thrown. Since time cannot be negative, the result $t = −1$ is discarded.

ⓑ The ball will be 80 feet above the ground when

h (t) = 80 .

This table details the step-by-step algebraic solution for determining when a projectile's height, h(t), reaches 80 feet, providing the equations and the interpretation of results.
	$h (t) = 80$
Substitute in the polynomial for $h (t) .$	$−16 t^{2} + 64 t + 80 = 80$
Subtract 80 from both sides.	$−16 t^{2} + 64 t = 0$
Factor the GCF, $−16 t .$	$−16 t (t - 4) = 0$
Use the Zero Product Property. Solve.	$\begin{matrix} −16 t & = & 0 & t - 4 & = & 0 \\ t & = & 0 & t & = & 4 \end{matrix}$
	The ball will be at 80 feet the moment Dennis tosses the ball and then 4 seconds later, when the ball is falling.

t = 2

seconds we find

h (2) .

This table demonstrates the step-by-step evaluation of the function h(t) = -16t^2 + 64t + 80 at t=2, determining the height of a ball.
	$h (t) = −16 t^{2} + 64 t + 80$
To find $h (2)$ substitute 2 for $t .$	$h (2) = −16 {(2)}^{2} + 64 \cdot 2 + 80$
Simplify.	$h (2) = 144$
	After 2 seconds, the ball will be at 144 feet.

Key Concepts

Polynomial Equation: A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.
Quadratic Equation: An equation of the form $a x^{2} + b x + c = 0$ is called a quadratic equation.

$a, b, c are real numbers and a \neq 0$
Zero Product Property: If $a \cdot b = 0,$ then either $a = 0$ or $b = 0$ or both.
How to use the Zero Product Property
1. Set each factor equal to zero.
2. Solve the linear equations.
3. Check.
How to solve a quadratic equation by factoring.
1. Write the quadratic equation in standard form, $a x^{2} + b x + c = 0 .$
2. Factor the quadratic expression.
3. Use the Zero Product Property.
4. Solve the linear equations.
5. Check. Substitute each solution separately into the original equation.
Zero of a Function: For any function f, if $f (x) = 0,$ then x is a zero of the function.
How to use a problem solving strategy to solve word problems.
1. Read the problem. Make sure all the words and ideas are understood.
2. Identify what we are looking for.
3. Name what we are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
5. Solve the equation using appropriate algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

Section Exercises

Practice Makes Perfect

Use the Zero Product Property

In the following exercises, solve.

$(3 a - 10) (2 a - 7) = 0$

Solution

$a = 10 / 3, a = 7 / 2$

$(5 b + 1) (6 b + 1) = 0$

$6 m (12 m - 5) = 0$

Solution

$m = 0, m = 5 / 12$

$2 x (6 x - 3) = 0$

${(2 x - 1)}^{2} = 0$

Solution

$x = 1 / 2$

${(3 y + 5)}^{2} = 0$

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$5 a^{2} - 26 a = 24$

Solution

$a = \frac{−4}{5}, a = 6$

$4 b^{2} + 7 b = −3$

$4 m^{2} = 17 m - 15$

Solution

$m = 5 / 4, m = 3$

$n^{2} = 5 n - 6$

$7 a^{2} + 14 a = 7 a$

Solution

$a = −1, a = 0$

$12 b^{2} - 15 b = −9 b$

$49 m^{2} = 144$

Solution

$m = 12 / 7, m = −12 / 7$

$625 = x^{2}$

$16 y^{2} = 81$

Solution

$y = −9 / 4, y = 9 / 4$

$64 p^{2} = 225$

$121 n^{2} = 36$

Solution

$n = −6 / 11, n = 6 / 11$

$100 y^{2} = 9$

$(x + 6) (x - 3) = −8$

Solution

$x = 2, x = −5$

$(p - 5) (p + 3) = −7$

$(2 x + 1) (x - 3) = −4 x$

Solution

$x = 3 / 2, x = −1$

$(y - 3) (y + 2) = 4 y$

$(3 x - 2) (x + 4) = 12 x$

Solution

$x = 2, x = −4 / 3$

$(2 y - 3) (3 y - 1) = 8 y$

$20 x^{2} - 60 x = −45$

Solution

$x = 3 / 2$

$3 y^{2} - 18 y = −27$

$15 x^{2} - 10 x = 40$

Solution

$x = 2, x = −4 / 3$

$14 y^{2} - 77 y = −35$

$18 x^{2} - 9 = −21 x$

Solution

$x = −3 / 2, x = 1 / 3$

$16 y^{2} + 12 = −32 y$

$16 p^{3} = 24 p^{2} - 9 p$

Solution

$p = 0, p = ¾$

$m^{3} - 2 m^{2} = − m$

$2 x^{3} + 72 x = 24 x^{2}$

Solution

$x = 0, x = 6$

$3 y^{3} + 48 y = 24 y^{2}$

$36 x^{3} + 24 x^{2} = −4 x$

Solution

$x = 0, x = –1 / 3$

$2 y^{3} + 2 y^{2} = 12 y$

Solve Equations with Polynomial Functions

In the following exercises, solve.

For the function, $f (x) = x^{2} - 8 x + 8,$ ⓐ find when $f (x) = −4$ ⓑ Use this information to find two points that lie on the graph of the function.

Solution

ⓐ $x = 2$ or $x = 6$ ⓑ $(2, −4)$ $(6, −4)$

For the function, $f (x) = x^{2} + 11 x + 20,$ ⓐ find when $f (x) = −8$ ⓑ Use this information to find two points that lie on the graph of the function.

For the function, $f (x) = 8 x^{2} - 18 x + 5,$ ⓐ find when $f (x) = −4$ ⓑ Use this information to find two points that lie on the graph of the function.

Solution

ⓐ $x = \frac{3}{2}$ or $x = \frac{3}{4}$
ⓑ $(\frac{3}{2}, −4)$ $(\frac{3}{4}, −4)$

For the function, $f (x) = 18 x^{2} + 15 x - 10,$ ⓐ find when $f (x) = 15$ ⓑ Use this information to find two points that lie on the graph of the function.

In the following exercises, for each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

$f (x) = 9 x^{2} - 4$

Solution

ⓐ $x = \frac{2}{3}$ or $x = - \frac{2}{3}$
ⓑ $(\frac{2}{3}, 0)$ , $(- \frac{2}{3}, 0)$ ⓒ $(0, −4)$

$f (x) = 25 x^{2} - 49$

$f (x) = 6 x^{2} - 7 x - 5$

Solution

ⓐ $x = \frac{5}{3}$ or $x = - \frac{1}{2}$
ⓑ $(\frac{5}{3}, 0)$ , $(- \frac{1}{2}, 0)$ ⓒ $(0, −5)$

$f (x) = 12 x^{2} - 11 x + 2$

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive odd integers is 143. Find the integers.

Solution

$−13, −11$ and 11, 13

The product of two consecutive odd integers is 195. Find the integers.

The product of two consecutive even integers is 168. Find the integers.

Solution

$−14, −12$ and 12, 14

The product of two consecutive even integers is 288. Find the integers.

The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.

Solution

Width: 4 feet; Length: 7 feet.

A rectangular retaining wall has area 15 square feet. The height of the wall is two feet less than its length. Find the height and the length of the wall.

The area of a bulletin board is 55 square feet. The length is four feet less than three times the width. Find the length and the width of the a bulletin board.

Solution

Width: 5 feet; Length: 11 feet.

A rectangular carport has area 150 square feet. The width of the carport is five feet less than twice its length. Find the width and the length of the carport.

A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.

Solution

The sides are 6 feet and 8 feet.

A stained glass window is shaped like a right triangle. The hypotenuse is 15 feet. One leg is three more than the other. Find the lengths of the legs.

A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is 9 feet longer than the side along the building. The third side is 7 feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.

Solution

The building side is 8 feet, the hypotenuse is 17 feet, and the third side is 15 feet.

A goat enclosure is in the shape of a right triangle. One leg of the enclosure is built against the side of the barn. The other leg is 4 feet more than the leg against the barn. The hypotenuse is 8 feet more than the leg along the barn. Find the three sides of the goat enclosure.

Juli is going to launch a model rocket in her back yard. When she launches the rocket, the function $h (t) = −16 t^{2} + 32 t$ models the height, h, of the rocket above the ground as a function of time, t. Find:

ⓐ the zeros of this function, which tell us when the rocket will be on the ground. ⓑ the time the rocket will be 16 feet above the ground.

Solution

ⓐ 0 seconds and 2 seconds ⓑ 1 second

Gianna is going to throw a ball from the top floor of her middle school. When she throws the ball from 48 feet above the ground, the function $h (t) = −16 t^{2} + 32 t + 48$ models the height, h, of the ball above the ground as a function of time, t. Find:

ⓐ the zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 48 feet above the ground. ⓒ the height the ball will be at $t = 1$ seconds which is when the ball will be at its highest point.

Writing Exercises

Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?

Solution

Answers will vary.

Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don’t get it. The first column has the following statements: solve quadratic equations by using the zero product property, solve quadratic equations by factoring and solve applications modeled by quadratic equations.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

Chapter Review Exercises

Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$12 a^{2} b^{3}, 15 a b^{2}$

Solution

$3 a b^{2}$

$12 m^{2} n^{3}, 42 m^{5} n^{3}$

$15 y^{3}, 21 y^{2}, 30 y$

Solution

$3 y$

$45 x^{3} y^{2}, 15 x^{4} y, 10 x^{5} y^{3}$

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$35 y + 84$

Solution

$7 (5 y + 12)$

$6 y^{2} + 12 y - 6$

$18 x^{3} - 15 x$

Solution

$3 x (6 x^{2} - 5)$

$15 m^{4} + 6 m^{2} n$

$4 x^{3} - 12 x^{2} + 16 x$

Solution

$4 x (x^{2} - 3 x + 4)$

$−3 x + 24$

$−3 x^{3} + 27 x^{2} - 12 x$

Solution

$−3 x (x^{2} - 9 x + 4)$

$3 x (x - 1) + 5 (x - 1)$

Factor by Grouping

In the following exercises, factor by grouping.

$a x - a y + b x - b y$

Solution

$(a + b) (x - y)$

$x^{2} y - x y^{2} + 2 x - 2 y$

$x^{2} + 7 x - 3 x - 21$

Solution

$(x - 3) (x + 7)$

$4 x^{2} - 16 x + 3 x - 12$

$m^{3} + m^{2} + m + 1$

Solution

$(m^{2} + 1) (m + 1)$

$5 x - 5 y - y + x$

Factor Trinomials

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c .$

$a^{2} + 14 a + 33$

Solution

$(a + 3) (a + 11)$

$k^{2} - 16 k + 60$

$m^{2} + 3 m - 54$

Solution

$(m + 9) (m - 6)$

$x^{2} - 3 x - 10$

In the following examples, factor each trinomial of the form $x^{2} + b x y + c y^{2} .$

$x^{2} + 12 x y + 35 y^{2}$

Solution

$(x + 5 y) (x + 7 y)$

$r^{2} + 3 r s - 28 s^{2}$

$a^{2} + 4 a b - 21 b^{2}$

Solution

$(a + 7 b) (a - 3 b)$

$p^{2} - 5 p q - 36 q^{2}$

$m^{2} - 5 m n + 30 n^{2}$

Solution

Prime

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

$x^{3} + 5 x^{2} - 24 x$

$3 y^{3} - 21 y^{2} + 30 y$

Solution

$3 y (y - 5) (y - 2)$

$5 x^{4} + 10 x^{3} - 75 x^{2}$

$5 y^{2} + 14 y + 9$

Solution

$(5 y + 9) (y + 1)$

$8 x^{2} + 25 x + 3$

$10 y^{2} - 53 y - 11$

Solution

$(5 y + 1) (2 y - 11)$

$6 p^{2} - 19 p q + 10 q^{2}$

$−81 a^{2} + 153 a + 18$

Solution

$−9 (9 a + 1) (a - 2)$

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor.

$2 x^{2} + 9 x + 4$

$18 a^{2} - 9 a + 1$

Solution

$(3 a - 1) (6 a - 1)$

$15 p^{2} + 2 p - 8$

$15 x^{2} + 6 x - 2$

Solution

Prime

$8 a^{2} + 32 a + 24$

$3 x^{2} + 3 x - 36$

Solution

$3 (x + 4) (x - 3)$

$48 y^{2} + 12 y - 36$

$18 a^{2} - 57 a - 21$

Solution

$3 (2 a - 7) (3 a + 1)$

$3 n^{4} - 12 n^{3} - 96 n^{2}$

Factor using substitution

In the following exercises, factor using substitution.

$x^{4} - 13 x^{2} - 30$

Solution

$(x^{2} - 15) (x^{2} + 2)$

${(x - 3)}^{2} - 5 (x - 3) - 36$

Factor Special Products

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

$25 x^{2} + 30 x + 9$

Solution

${(5 x + 3)}^{2}$

$36 a^{2} - 84 a b + 49 b^{2}$

$40 x^{2} + 360 x + 810$

Solution

$10 {(2 x + 9)}^{2}$

$5 k^{3} - 70 k^{2} + 245 k$

$75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2}$

Solution

$3 u^{2} {(5 u - v)}^{2}$

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$81 r^{2} - 25$

$169 m^{2} - n^{2}$

Solution

$(13 m + n) (13 m - n)$

$25 p^{2} - 1$

$9 - 121 y^{2}$

Solution

$(3 + 11 y) (3 - 11 y)$

$20 x^{2} - 125$

$169 n^{3} - n$

Solution

$n (13 n + 1) (13 n - 1)$

$6 p^{2} q^{2} - 54 p^{2}$

$24 p^{2} + 54$

Solution

$6 (4 p^{2} + 9)$

$49 x^{2} - 81 y^{2}$

$16 z^{4} - 1$

Solution

$(2 z - 1) (2 z + 1) (4 z^{2} + 1)$

$48 m^{4} n^{2} - 243 n^{2}$

$a^{2} + 6 a + 9 - 9 b^{2}$

Solution

$(a + 3 - 3 b) (a + 3 + 3 b)$

$x^{2} - 16 x + 64 - y^{2}$

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$a^{3} - 125$

Solution

$(a - 5) (a^{2} + 5 a + 25)$

$b^{3} - 216$

$2 m^{3} + 54$

Solution

$2 (m + 3) (m^{2} - 3 m + 9)$

$81 m^{3} + 3$

General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$24 x^{3} + 44 x^{2}$

Solution

$4 x^{2} (6 x + 11)$

$24 a^{4} - 9 a^{3}$

$16 n^{2} - 56 m n + 49 m^{2}$

Solution

${(4 n - 7 m)}^{2}$

$6 a^{2} - 25 a - 9$

$5 u^{4} - 45 u^{2}$

Solution

$5 u^{2} (u + 3) (u - 3)$

$n^{4} - 81$

$64 j^{2} + 225$

Solution

prime

$5 x^{2} + 5 x - 60$

$b^{3} - 64$

Solution

$(b - 4) (b^{2} + 4 b + 16)$

$m^{3} + 125$

$2 b^{2} - 2 b c + 5 c b - 5 c^{2}$

Solution

$(2 b + 5 c) (b - c)$

$48 x^{5} y^{2} - 243 x y^{2}$

$5 q^{2} - 15 q - 90$

Solution

$5 (q + 3) (q - 6)$

$4 u^{5} v + 4 u^{2} v^{3}$

$10 m^{4} - 6250$

Solution

$10 (m - 5) (m + 5) (m^{2} + 25)$

$60 x^{2} y - 75 x y + 30 y$

$16 x^{2} - 24 x y + 9 y^{2} - 64$

Solution

$(4 x - 3 y + 8) (4 x - 3 y - 8)$

Polynomial Equations

Use the Zero Product Property

In the following exercises, solve.

$(a - 3) (a + 7) = 0$

$(5 b + 1) (6 b + 1) = 0$

Solution

$b = −1 / 5, b = −1 / 6$

$6 m (12 m - 5) = 0$

${(2 x - 1)}^{2} = 0$

Solution

$x = 1 / 2$

$3 m (2 m - 5) (m + 6) = 0$

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$x^{2} + 9 x + 20 = 0$

Solution

$x = −4, x = −5$

$y^{2} - y - 72 = 0$

$2 p^{2} - 11 p = 40$

Solution

$p = - \frac{5}{2}, p = 8$

$q^{3} + 3 q^{2} + 2 q = 0$

$144 m^{2} - 25 = 0$

Solution

$m = \frac{5}{12}, m = - \frac{5}{12}$

$4 n^{2} = 36$

$(x + 6) (x - 3) = −8$

Solution

$x = 2, x = −5$

$(3 x - 2) (x + 4) = 12 x$

$16 p^{3} = 24 p^{2} - 9 p$

Solution

$p = 0, p = ¾$

$2 y^{3} + 2 y^{2} = 12 y$

Solve Equations with Polynomial Functions

In the following exercises, solve.

For the function, $f (x) = x^{2} + 11 x + 20,$ ⓐ find when $f (x) = −8$ ⓑ Use this information to find two points that lie on the graph of the function.

Solution

ⓐ $x = −7$ or $x = −4$
ⓑ $(−7, −8)$ $(−4, −8)$

For the function, $f (x) = 9 x^{2} - 18 x + 5,$ ⓐ find when $f (x) = −3$ ⓑ Use this information to find two points that lie on the graph of the function.

In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

$f (x) = 64 x^{2} - 49$

Solution

$f (x) = 6 x^{2} - 13 x - 5$

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive odd numbers is 399. Find the numbers.

Solution

The numbers are $−21$ and $−19$ or 19 and 21.

The area of a rectangular shaped patio 432 square feet. The length of the patio is 6 feet more than its width. Find the length and width.

A ladder leans against the wall of a building. The length of the ladder is 9 feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is 7 feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning against the building.

Solution

The lengths are 8, 15, and 17 ft.

Shruti is going to throw a ball from the top of a cliff. When she throws the ball from 80 feet above the ground, the function $h (t) = −16 t^{2} + 64 t + 80$ models the height, h, of the ball above the ground as a function of time, t. Find: ⓐ the zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 80 feet above the ground. ⓒ the height the ball will be at $t = 2$ seconds which is when the ball will be at its highest point.

Chapter Practice Test

In the following exercises, factor completely.

$80 a^{2} + 120 a^{3}$

Solution

$40 a^{2} (2 + 3 a)$

$5 m (m - 1) + 3 (m - 1)$

$x^{2} + 13 x + 36$

Solution

$(x + 4) (x + 9)$

$p^{2} + p q - 12 q^{2}$

$x y - 8 y + 7 x - 56$

Solution

$(x - 8) (y + 7)$

$40 r^{2} + 810$

$9 s^{2} - 12 s + 4$

Solution

${(3 s - 2)}^{2}$

$6 x^{2} - 11 x - 10$

$3 x^{2} - 75 y^{2}$

Solution

$3 (x + 5 y) (x - 5 y)$

$6 u^{2} + 3 u - 18$

$x^{3} + 125$

Solution

$(x + 5) (x^{2} - 5 x + 25)$

$32 x^{5} y^{2} - 162 x y^{2}$

$6 x^{4} - 19 x^{2} + 15$

Solution

$(3 x^{2} - 5) (2 x^{2} - 3)$

$3 x^{3} - 36 x^{2} + 108 x$

In the following exercises, solve

$5 a^{2} + 26 a = 24$

Solution

$a = 4 / 5, a = −6$

The product of two consecutive integers is 156. Find the integers.

The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.

Solution

The width is 12 inches and the length is 14 inches.

Jing is going to throw a ball from the balcony of her condo. When she throws the ball from 80 feet above the ground, the function $h (t) = −16 t^{2} + 64 t + 80$ models the height, h, of the ball above the ground as a function of time, t. Find: ⓐ the zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 128 feet above the ground. ⓒ the height the ball will be at $t = 4$ seconds.

For the function, $f (x) = x^{2} - 7 x + 5,$ ⓐ find when $f (x) = −7$ ⓑ Use this information to find two points that lie on the graph of the function.

Solution

ⓐ $x = 3$ or $x = 4$ ⓑ $(3, −7)$ $(4, −7)$

For the function $f (x) = 25 x^{2} - 81,$ find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

degree of the polynomial equation

The degree of the polynomial equation is the degree of the polynomial.

polynomial equation

A polynomial equation is an equation that contains a polynomial expression.

quadratic equation

Polynomial equations of degree two are called quadratic equations.

zero of the function

A value of

x

where the function is 0, is called a zero of the function.

Zero Product Property

The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero.