Chapter 6 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

degree of the polynomial equation

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

factoring

Splitting a product into factors is called factoring.

greatest common factor

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

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

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.

##

Key Concepts

• How to find the greatest common factor (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.

• Distributive Property: If a, b, and c are real numbers, then

The form on the left is used to multiply. The form on the right is used to factor.

• How to factor the 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.

• Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.

• How to factor by grouping. 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.

• How to factor trinomials of the form ${x}^{2}+bx+c.$ Step 1. Write the factors as two binomials with first terms x. $\,\begin{array}{c} {x}^{2}+bx+c \\ (x\,)(x\,) \\ \end{array}$ Step 2. Find two numbers m and n that $\begin{array}{lcl} \mathrm{multiply to} & & c,m\cdot{}n=c \\ \mathrm{add to} & & b,m+n=b \\ \end{array}$ Step 3. Use m and n as the last terms of the factors. $\,(x+m)(x+n)$ Step 4. Check by multiplying the factors.

• Strategy for Factoring Trinomials of the Form ${x}^{2}+bx+c$: When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors. $\begin{array}{c} \begin{array}{c} {x}^{2}+bx+c \\ (x+m)(x+n) \\ \end{array} \\ \mathrm{When}\,c\,\mathrm{is positive,}\,m\,\mathrm{and}\,n\,\mathrm{have the same sign.} \\ b\,\mathrm{positive}\,b\,\mathrm{negative} \\ m,n\,\mathrm{positive}\,m,n\,\mathrm{negative} \\ {x}^{2}+5x+6\,{x}^{2}-6x+8 \\ (x+2)(x+3)\,(x-4)(x-2) \\ \mathrm{same signs}\,\mathrm{same signs} \\ \mathrm{When}\,c\,\mathrm{is negative,}\,m\,\mathrm{and}\,n\,\mathrm{have opposite signs.} \\ {x}^{2}+x-12\,{x}^{2}-2x-15 \\ (x+4)(x-3)\,(x-5)(x+3) \\ \mathrm{opposite signs}\,\mathrm{opposite signs} \\ \end{array}$ Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.

• How to factor trinomials of the form $a{x}^{2}+bx+c$ using trial and error. Step 1. Write the trinomial in descending order of degrees as needed. Step 2. Factor any GCF. Step 3. Find all the factor pairs of the first term. Step 4. Find all the factor pairs of the third term. Step 5. Test all the possible combinations of the factors until the correct product is found. Step 6. Check by multiplying.

• How to factor trinomials of the form $a{x}^{2}+bx+c$ using the “ac” method. Step 1. Factor any GCF. Step 2. Find the product ac. Step 3. Find two numbers m and n that: $\begin{array}{lccccl} \mathrm{Multiply to}\,ac. & & & & & m\cdot{}n=a\cdot{}c \\ \mathrm{Add to}\,b. & & & & & m+n=b \\ & & & & & a{x}^{2}+bx+c \\ \end{array}$ Step 4. Split the middle term using m and n. $\,a{x}^{2}+mx+nx+c$ Step 5. Factor by grouping. Step 6. Check by multiplying the factors.

• Perfect Square Trinomials Pattern: If a and b are real numbers,

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

• Difference of Squares Pattern: If $a,b$ are real numbers,

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

• Sum and Difference of Cubes Pattern $\begin{array}{c} {a}^{3}+{b}^{3}=(a+b)({a}^{2}-ab+{b}^{2}) \\ {a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2}) \\ \end{array}$

• How to factor the sum or difference of cubes. 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.

• How to use a general strategy for factoring 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?

• 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 $a{x}^{2}+bx+c=0$ is called a quadratic equation.

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

• How to use the Zero Product Property Step 1. Set each factor equal to zero. Step 2. Solve the linear equations. Step 3. Check.

• How to solve a quadratic equation by factoring. 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. 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. 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 algebraic equation. Step 5. Solve the equation using appropriate 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 ${x}^{2}+bx+c.$

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

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

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

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

In the following exercises, factor.

Factor using substitution

In the following exercises, factor using substitution.

Factor Perfect Square Trinomials

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

Factor Differences of Squares

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

Factor Sums and Differences of Cubes

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

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 Equations with Polynomial Functions

In the following exercises, solve.

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.

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

##

In the following exercises, factor completely.

In the following exercises, solve

Key Terms

degree of the polynomial equation

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

factoring

Splitting a product into factors is called factoring.

greatest common factor

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

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

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.

##

Key Concepts

• How to find the greatest common factor (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.

• Distributive Property: If a, b, and c are real numbers, then

The form on the left is used to multiply. The form on the right is used to factor.

• How to factor the 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.

• Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.

• How to factor by grouping. 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.

• How to factor trinomials of the form ${x}^{2}+bx+c.$ Step 1. Write the factors as two binomials with first terms x. $\,\begin{array}{c} {x}^{2}+bx+c \\ (x\,)(x\,) \\ \end{array}$ Step 2. Find two numbers m and n that $\begin{array}{lcl} \mathrm{multiply to} & & c,m\cdot{}n=c \\ \mathrm{add to} & & b,m+n=b \\ \end{array}$ Step 3. Use m and n as the last terms of the factors. $\,(x+m)(x+n)$ Step 4. Check by multiplying the factors.

• Strategy for Factoring Trinomials of the Form ${x}^{2}+bx+c$: When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors. $\begin{array}{c} \begin{array}{c} {x}^{2}+bx+c \\ (x+m)(x+n) \\ \end{array} \\ \mathrm{When}\,c\,\mathrm{is positive,}\,m\,\mathrm{and}\,n\,\mathrm{have the same sign.} \\ b\,\mathrm{positive}\,b\,\mathrm{negative} \\ m,n\,\mathrm{positive}\,m,n\,\mathrm{negative} \\ {x}^{2}+5x+6\,{x}^{2}-6x+8 \\ (x+2)(x+3)\,(x-4)(x-2) \\ \mathrm{same signs}\,\mathrm{same signs} \\ \mathrm{When}\,c\,\mathrm{is negative,}\,m\,\mathrm{and}\,n\,\mathrm{have opposite signs.} \\ {x}^{2}+x-12\,{x}^{2}-2x-15 \\ (x+4)(x-3)\,(x-5)(x+3) \\ \mathrm{opposite signs}\,\mathrm{opposite signs} \\ \end{array}$ Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.

• How to factor trinomials of the form $a{x}^{2}+bx+c$ using trial and error. Step 1. Write the trinomial in descending order of degrees as needed. Step 2. Factor any GCF. Step 3. Find all the factor pairs of the first term. Step 4. Find all the factor pairs of the third term. Step 5. Test all the possible combinations of the factors until the correct product is found. Step 6. Check by multiplying.

• How to factor trinomials of the form $a{x}^{2}+bx+c$ using the “ac” method. Step 1. Factor any GCF. Step 2. Find the product ac. Step 3. Find two numbers m and n that: $\begin{array}{lccccl} \mathrm{Multiply to}\,ac. & & & & & m\cdot{}n=a\cdot{}c \\ \mathrm{Add to}\,b. & & & & & m+n=b \\ & & & & & a{x}^{2}+bx+c \\ \end{array}$ Step 4. Split the middle term using m and n. $\,a{x}^{2}+mx+nx+c$ Step 5. Factor by grouping. Step 6. Check by multiplying the factors.

• Perfect Square Trinomials Pattern: If a and b are real numbers,

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

• Difference of Squares Pattern: If $a,b$ are real numbers,

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

• Sum and Difference of Cubes Pattern $\begin{array}{c} {a}^{3}+{b}^{3}=(a+b)({a}^{2}-ab+{b}^{2}) \\ {a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2}) \\ \end{array}$

• How to factor the sum or difference of cubes. 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.

• How to use a general strategy for factoring 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?

• 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 $a{x}^{2}+bx+c=0$ is called a quadratic equation.

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

• How to use the Zero Product Property Step 1. Set each factor equal to zero. Step 2. Solve the linear equations. Step 3. Check.

• How to solve a quadratic equation by factoring. 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. 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. 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 algebraic equation. Step 5. Solve the equation using appropriate 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 ${x}^{2}+bx+c.$

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

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

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

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

In the following exercises, factor.

Factor using substitution

In the following exercises, factor using substitution.

Factor Perfect Square Trinomials

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

Factor Differences of Squares

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

Factor Sums and Differences of Cubes

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

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 Equations with Polynomial Functions

In the following exercises, solve.

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.

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

##

In the following exercises, factor completely.

In the following exercises, solve

1. $12{a}^{2}{b}^{3},15a{b}^{2}$

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

3. $35y+84$

4. $18{x}^{3}-15x$

1. $4{x}^{3}-12{x}^{2}+16x$

2. $-3{x}^{3}+27{x}^{2}-12x$

3. $ax-ay+bx-by$

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

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

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

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

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

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

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

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

4. $5{y}^{2}+14y+9$

1. $10{y}^{2}-53y-11$

2. $-81{a}^{2}+153a+18$

3. $18{a}^{2}-9a+1$

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

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

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

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

4. $25{x}^{2}+30x+9$

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

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

3. $169{m}^{2}-{n}^{2}$

4. $9-121{y}^{2}$

1. $169{n}^{3}-n$

2. $24{p}^{2}+54$

3. $16{z}^{4}-1$

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

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

2. $64{j}^{2}+225$

3. ${b}^{3}-64$

4. $2{b}^{2}-2bc+5cb-5{c}^{2}$

1. $5{q}^{2}-15q-90$

2. $10{m}^{4}-6250$

3. $16{x}^{2}-24xy+9{y}^{2}-64$

4. $(5b+1)(6b+1)=0$

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

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

3. $2{p}^{2}-11p=40$

4. $144{m}^{2}-25=0$

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

2. 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.

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

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

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

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

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

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

1. $(x+6)(x-3)=-8$

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

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

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

1. $6{x}^{4}-19{x}^{2}+15$

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

3. 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.

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