Chapter 7 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

##

Worked example

difference of squares pattern

a

b

factoring

Factoring is splitting a product into factors; in other words, it is the reverse process of multiplying.

greatest common factor

The greatest common factor is the largest expression that is a factor of two or more expressions is the greatest common factor (GCF).

perfect square trinomials pattern

a

b

prime polynomials

Polynomials that cannot be factored are prime polynomials.

quadratic equations

are equations in which the variable is squared.

sum and difference of cubes pattern

Zero Product Property

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

##

Key Concepts

• Finding the Greatest Common Factor (GCF): To find the GCF of two expressions: Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors. Step 3. Bring down the common factors that all expressions share. Step 4. Multiply the factors as in Example 7.2.

• Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial: Step 1. Find the GCF of all the terms of the polynomial. Step 2. Rewrite each term as a product using the GCF. Step 3. Use the ‘reverse’ Distributive Property to factor the expression. Step 4. Check by multiplying the factors as in Example 7.5.

• Factor by Grouping: To factor a polynomial with 4 four or more terms Step 1. Group terms with common factors. Step 2. Factor out the common factor in each group. Step 3. Factor the common factor from the expression. Step 4. Check by multiplying the factors as in Example 7.15.

• Factor trinomials of the form ${x}^{2}+bx+c$ Step 1. Write the factors as two binomials with first terms x: $(x\,)(x\,)$. Step 2. Find two numbers m and n that Multiply to c, $m\cdot{}n=c$ Add to b, $m+n=b$ Step 3. Use m and n as the last terms of the factors: $(x+m)(x+n)$. Step 4. Check by multiplying the factors.

• Factor Trinomials of the Form $a{x}^{2}+bx+c$ using Trial and Error: See Example 7.33. Step 1. Write the trinomial in descending order of degrees. Step 2. Find all the factor pairs of the first term. Step 3. Find all the factor pairs of the third term. Step 4. Test all the possible combinations of the factors until the correct product is found. Step 5. Check by multiplying.

• Factor Trinomials of the Form $a{x}^{2}+bx+c$ Using the “ac” Method: See Example 7.38. Step 1. Factor any GCF. Step 2. Find the product ac. Step 3. Find two numbers m and n that: $\begin{array}{lccl} \mathrm{Multiply to}\,ac & & & m\cdot{}n=a\cdot{}c \\ \mathrm{Add to}\,b & & & m+n=b \\ \end{array}$ Step 4. Split the middle term using m and n: Step 5. Factor by grouping. Step 6. Check by multiplying the factors.

• Choose a strategy to factor polynomials completely (updated): Step 1. Is there a greatest common factor? Factor it. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms? If it is a binomial, right now we have no method to factor it. If it is a trinomial of the form ${x}^{2}+bx+c$ Undo FOIL $(x\,)(x\,)$. If it is a trinomial of the form $a{x}^{2}+bx+c$ Use Trial and Error or the “ac” method. If it has more than three terms Use the grouping method. Step 3. Check by multiplying the factors.

• Factor perfect square trinomials See Example 7.42. $\begin{array}{lcccccc} \mathrm{Step 1.}\,\mathrm{Does the trinomial fit the pattern?} & & & {a}^{2}+2ab+{b}^{2} & & & {a}^{2}-2ab+{b}^{2} \\ \,\mathrm{Is the first term a perfect square?} & & & {(a)}^{2} & & & {(a)}^{2} \\ \,\mathrm{Write it as a square.} & & & & & & \\ \,\mathrm{Is the last term a perfect square?} & & & {(a)}^{2}\,{(b)}^{2} & & & {(a)}^{2}\,{(b)}^{2} \\ \,\mathrm{Write it as a square.} & & & & & & \\ \,\mathrm{Check the middle term. Is it}\,2ab? & & & {(a)}^{2}{}_{↘}\underset{2\cdot{}a\cdot{}b}{}{}_{↙}{(b)}^{2} & & & {(a)}^{2}{}_{↘}\underset{2\cdot{}a\cdot{}b}{}{}_{↙}{(b)}^{2} \\ \mathrm{Step 2.}\,\mathrm{Write the square of the binomial.} & & & {(a+b)}^{2} & & & {(a-b)}^{2} \\ \mathrm{Step 3.}\,\mathrm{Check by multiplying.} & & & & & & \\ \end{array}$

• Factor differences of squares See Example 7.47. $\begin{array}{lccc} \mathrm{Step 1.}\,\mathrm{Does the binomial fit the pattern?} & & & {a}^{2}-{b}^{2} \\ \,\mathrm{Is this a difference?} & & & \underline{\hspace{2em}}-\underline{\hspace{2em}} \\ \,\mathrm{Are the first and last terms perfect squares?} & & & \\ \mathrm{Step 2.}\,\mathrm{Write them as squares.} & & & {(a)}^{2}-{(b)}^{2} \\ \mathrm{Step 3.}\,\mathrm{Write the product of conjugates.} & & & (a-b)(a+b) \\ \mathrm{Step 4.}\,\mathrm{Check by multiplying.} & & & \\ \end{array}$

• Factor sum and difference of cubes To factor the sum or difference of cubes: See Example 7.54. Step 1. Does the binomial fit the sum or difference of cubes pattern? Is it a sum or difference? Are the first and last terms perfect cubes? Step 2. Write them as cubes. Step 3. Use either the sum or difference of cubes pattern. Step 4. Simplify inside the parentheses Step 5. Check by multiplying the factors.

• General Strategy for Factoring Polynomials See Figure 7.3.

• How to Factor Polynomials Step 1. Is there a greatest common factor? Factor it out. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms? If it is a binomial: Is it a sum? Of squares? Sums of squares do not factor. Of cubes? Use the sum of cubes pattern. Is it a difference? Of squares? Factor as the product of conjugates. Of cubes? Use the difference of cubes pattern. If it is a trinomial: Is it of the form ${x}^{2}+bx+c$? Undo FOIL. Is it of the form $a{x}^{2}+bx+c$? If ‘a’ and ‘c’ are squares, check if it fits the trinomial square pattern. Use the trial and error or ‘ac’ method. If it has more than three terms: Use the grouping method. Step 3. Check. Is it factored completely? Do the factors multiply back to the original polynomial?

• Zero Product Property If $a\cdot{}b=0$, then either $a=0$ or $b=0$ or both. See Example 7.69.

• Solve a quadratic equation by factoring To solve a quadratic equation by factoring: See Example 7.73. Step 1. Write the quadratic equation in standard form, $a{x}^{2}+bx+c=0$. Step 2. Factor the quadratic expression. Step 3. Use the Zero Product Property. Step 4. Solve the linear equations. Step 5. Check.

• Use a problem solving strategy to solve word problems See Example 7.80. Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 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 algebra equation. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

##

Review Exercises

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

Factor the Greatest Common Factor from a Polynomial

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

Factor by Grouping

In the following exercises, factor by grouping.

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

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

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

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

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

Factor Trinomials of the Form $a{x}^{2}+bx+c$ with a GCF

In the following exercises, factor completely.

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

Factor Trinomials with a GCF Using the “ac” Method

In the following exercises, factor.

Factor Perfect Square Trinomials

In the following exercises, factor.

Factor Differences of Squares

In the following exercises, factor.

Factor Sums and Differences of Cubes

In the following exercises, factor.

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

Use the Zero Product Property

In the following exercises, solve.

Solve Quadratic Equations by Factoring

In the following exercises, solve.

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

##

In the following exercises, find the Greatest Common Factor in each expression.

In the following exercises, factor completely.

In the following exercises, solve.

Key Terms

difference of squares pattern

a

b

factoring

Factoring is splitting a product into factors; in other words, it is the reverse process of multiplying.

greatest common factor

The greatest common factor is the largest expression that is a factor of two or more expressions is the greatest common factor (GCF).

perfect square trinomials pattern

a

b

prime polynomials

Polynomials that cannot be factored are prime polynomials.

quadratic equations

are equations in which the variable is squared.

sum and difference of cubes pattern

Zero Product Property

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

##

Key Concepts

• Finding the Greatest Common Factor (GCF): To find the GCF of two expressions: Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors. Step 3. Bring down the common factors that all expressions share. Step 4. Multiply the factors as in Example 7.2.

• Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial: Step 1. Find the GCF of all the terms of the polynomial. Step 2. Rewrite each term as a product using the GCF. Step 3. Use the ‘reverse’ Distributive Property to factor the expression. Step 4. Check by multiplying the factors as in Example 7.5.

• Factor by Grouping: To factor a polynomial with 4 four or more terms Step 1. Group terms with common factors. Step 2. Factor out the common factor in each group. Step 3. Factor the common factor from the expression. Step 4. Check by multiplying the factors as in Example 7.15.

• Factor trinomials of the form ${x}^{2}+bx+c$ Step 1. Write the factors as two binomials with first terms x: $(x\,)(x\,)$. Step 2. Find two numbers m and n that Multiply to c, $m\cdot{}n=c$ Add to b, $m+n=b$ Step 3. Use m and n as the last terms of the factors: $(x+m)(x+n)$. Step 4. Check by multiplying the factors.

• Factor Trinomials of the Form $a{x}^{2}+bx+c$ using Trial and Error: See Example 7.33. Step 1. Write the trinomial in descending order of degrees. Step 2. Find all the factor pairs of the first term. Step 3. Find all the factor pairs of the third term. Step 4. Test all the possible combinations of the factors until the correct product is found. Step 5. Check by multiplying.

• Factor Trinomials of the Form $a{x}^{2}+bx+c$ Using the “ac” Method: See Example 7.38. Step 1. Factor any GCF. Step 2. Find the product ac. Step 3. Find two numbers m and n that: $\begin{array}{lccl} \mathrm{Multiply to}\,ac & & & m\cdot{}n=a\cdot{}c \\ \mathrm{Add to}\,b & & & m+n=b \\ \end{array}$ Step 4. Split the middle term using m and n: Step 5. Factor by grouping. Step 6. Check by multiplying the factors.

• Choose a strategy to factor polynomials completely (updated): Step 1. Is there a greatest common factor? Factor it. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms? If it is a binomial, right now we have no method to factor it. If it is a trinomial of the form ${x}^{2}+bx+c$ Undo FOIL $(x\,)(x\,)$. If it is a trinomial of the form $a{x}^{2}+bx+c$ Use Trial and Error or the “ac” method. If it has more than three terms Use the grouping method. Step 3. Check by multiplying the factors.

• Factor perfect square trinomials See Example 7.42. $\begin{array}{lcccccc} \mathrm{Step 1.}\,\mathrm{Does the trinomial fit the pattern?} & & & {a}^{2}+2ab+{b}^{2} & & & {a}^{2}-2ab+{b}^{2} \\ \,\mathrm{Is the first term a perfect square?} & & & {(a)}^{2} & & & {(a)}^{2} \\ \,\mathrm{Write it as a square.} & & & & & & \\ \,\mathrm{Is the last term a perfect square?} & & & {(a)}^{2}\,{(b)}^{2} & & & {(a)}^{2}\,{(b)}^{2} \\ \,\mathrm{Write it as a square.} & & & & & & \\ \,\mathrm{Check the middle term. Is it}\,2ab? & & & {(a)}^{2}{}_{↘}\underset{2\cdot{}a\cdot{}b}{}{}_{↙}{(b)}^{2} & & & {(a)}^{2}{}_{↘}\underset{2\cdot{}a\cdot{}b}{}{}_{↙}{(b)}^{2} \\ \mathrm{Step 2.}\,\mathrm{Write the square of the binomial.} & & & {(a+b)}^{2} & & & {(a-b)}^{2} \\ \mathrm{Step 3.}\,\mathrm{Check by multiplying.} & & & & & & \\ \end{array}$

• Factor differences of squares See Example 7.47. $\begin{array}{lccc} \mathrm{Step 1.}\,\mathrm{Does the binomial fit the pattern?} & & & {a}^{2}-{b}^{2} \\ \,\mathrm{Is this a difference?} & & & \underline{\hspace{2em}}-\underline{\hspace{2em}} \\ \,\mathrm{Are the first and last terms perfect squares?} & & & \\ \mathrm{Step 2.}\,\mathrm{Write them as squares.} & & & {(a)}^{2}-{(b)}^{2} \\ \mathrm{Step 3.}\,\mathrm{Write the product of conjugates.} & & & (a-b)(a+b) \\ \mathrm{Step 4.}\,\mathrm{Check by multiplying.} & & & \\ \end{array}$

• Factor sum and difference of cubes To factor the sum or difference of cubes: See Example 7.54. Step 1. Does the binomial fit the sum or difference of cubes pattern? Is it a sum or difference? Are the first and last terms perfect cubes? Step 2. Write them as cubes. Step 3. Use either the sum or difference of cubes pattern. Step 4. Simplify inside the parentheses Step 5. Check by multiplying the factors.

• General Strategy for Factoring Polynomials See Figure 7.3.

• How to Factor Polynomials Step 1. Is there a greatest common factor? Factor it out. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms? If it is a binomial: Is it a sum? Of squares? Sums of squares do not factor. Of cubes? Use the sum of cubes pattern. Is it a difference? Of squares? Factor as the product of conjugates. Of cubes? Use the difference of cubes pattern. If it is a trinomial: Is it of the form ${x}^{2}+bx+c$? Undo FOIL. Is it of the form $a{x}^{2}+bx+c$? If ‘a’ and ‘c’ are squares, check if it fits the trinomial square pattern. Use the trial and error or ‘ac’ method. If it has more than three terms: Use the grouping method. Step 3. Check. Is it factored completely? Do the factors multiply back to the original polynomial?

• Zero Product Property If $a\cdot{}b=0$, then either $a=0$ or $b=0$ or both. See Example 7.69.

• Solve a quadratic equation by factoring To solve a quadratic equation by factoring: See Example 7.73. Step 1. Write the quadratic equation in standard form, $a{x}^{2}+bx+c=0$. Step 2. Factor the quadratic expression. Step 3. Use the Zero Product Property. Step 4. Solve the linear equations. Step 5. Check.

• Use a problem solving strategy to solve word problems See Example 7.80. Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 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 algebra equation. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

##

Review Exercises

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

Factor the Greatest Common Factor from a Polynomial

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

Factor by Grouping

In the following exercises, factor by grouping.

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

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

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

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

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

Factor Trinomials of the Form $a{x}^{2}+bx+c$ with a GCF

In the following exercises, factor completely.

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

Factor Trinomials with a GCF Using the “ac” Method

In the following exercises, factor.

Factor Perfect Square Trinomials

In the following exercises, factor.

Factor Differences of Squares

In the following exercises, factor.

Factor Sums and Differences of Cubes

In the following exercises, factor.

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

Use the Zero Product Property

In the following exercises, solve.

Solve Quadratic Equations by Factoring

In the following exercises, solve.

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

##

In the following exercises, find the Greatest Common Factor in each expression.

In the following exercises, factor completely.

In the following exercises, solve.

1. $90,150,105$

2. $24x-42$

3. $15{m}^{4}+6{m}^{2}n$

4. $ax-ay+bx-by$

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

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

3. ${u}^{2}+17u+72$

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

1. ${y}^{2}+6y-7$

2. ${s}^{2}-2s-8$

3. ${x}^{2}+12xy+35{y}^{2}$

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

1. ${y}^{2}-17y+42$

2. $8{a}^{3}+72a$

3. $6{x}^{2}+42x+60$

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

1. $2{x}^{2}+9x+4$

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

3. $15{p}^{2}+2p-8$

4. $40{s}^{2}-s-6$

1. $3{x}^{2}+3x-36$

2. $60{y}^{2}-85y-25$

3. $25{x}^{2}+30x+9$

4. $36{a}^{2}-84ab+49{b}^{2}$

1. $40{x}^{2}+360x+810$

2. $2{y}^{3}-16{y}^{2}+32y$

3. $81{r}^{2}-25$

4. $169{m}^{2}-{n}^{2}$

1. $25{p}^{2}-1$

2. $9-121{y}^{2}$

3. $20{x}^{2}-125$

4. $49{u}^{3}-9u$

1. ${a}^{3}-125$

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

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

4. $16{n}^{2}-56mn+49{m}^{2}$

1. $5{r}^{2}+22r-48$

2. ${n}^{4}-81$

3. $5{x}^{2}+5x-60$

4. ${m}^{3}+125$

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

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

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

4. $2{p}^{2}-11p=40$

1. $144{m}^{2}-25=0$

2. The product of two consecutive numbers is $462$. Find the numbers.

3. $14y-42$

4. $80{a}^{2}+120{a}^{3}$

1. ${x}^{2}+13x+42$

2. $3{a}^{3}-6{a}^{2}-72a$

3. $5{n}^{2}+30n+45$

4. $xy-8y+7x-56$

1. $9{s}^{2}-12s+4$

2. $100-{a}^{2}$

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

4. $ab-3b-2a+6$

1. $8{m}^{2}+22m+5$

2. ${y}^{2}=y+132$

3. $9{b}^{2}-9=0$

4. $4{n}^{2}+19n+21=0$

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