Chapter 5 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.

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Worked example

binomial

A binomial is a polynomial with exactly two terms.

conjugate pair

degree of a constant

The degree of any constant is 0.

degree of a polynomial

The degree of a polynomial is the highest degree of all its terms.

degree of a term

The degree of a term is the sum of the exponents of its variables.

monomial

a

m

polynomial

A monomial or two or more monomials combined by addition or subtraction is a polynomial.

polynomial function

A polynomial function is a function whose range values are defined by a polynomial.

Power Property

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n

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n

Product Property

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Product to a Power

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Properties of Negative Exponents

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Quotient Property

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n

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Quotient to a Negative Exponent

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Quotient to a Power Property

a

b

m

a

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b

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b

standard form of a polynomial

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.

trinomial

A trinomial is a polynomial with exactly three terms.

Zero Exponent Property

a

a

##

Key Concepts

• Monomial A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form $a{x}^{m},$ where a is a constant and m is a whole number.

• Polynomials Polynomial—A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial. monomial —A polynomial with exactly one term is called a monomial. binomial — A polynomial with exactly two terms is called a binomial. trinomial —A polynomial with exactly three terms is called a trinomial.

• Degree of a Polynomial The degree of a term is the sum of the exponents of its variables. The degree of a constant is 0. The degree of a polynomial is the highest degree of all its terms.

• Exponential Notation This is read a to the ${m}^{th}$ power. In the expression ${a}^{m}$, the exponent m tells us how many times we use the base a as a factor.

• Product Property for Exponents If a is a real number and m and n are integers, then

To multiply with like bases, add the exponents.

• Quotient Property for Exponents If $a$ is a real number, $a\ne{}0,$ and m and n are integers, then

• Zero Exponent If a is a non-zero number, then ${a}^{0}=1.$ If a is a non-zero number, then a to the power of zero equals 1. Any non-zero number raised to the zero power is 1.

• Negative Exponent If n is an integer and $a\ne{}0,$ then ${a}^{-n}=\frac{1}{{a}^{n}}$ or $\frac{1}{{a}^{-n}}={a}^{n}.$

• Quotient to a Negative Exponent Property If $a,b$ are real numbers, $a\ne{}0,b\ne{}0$ and $n$ is an integer, then

• Power Property for Exponents If $a$ is a real number and $m,n$ are integers, then

To raise a power to a power, multiply the exponents.

• Product to a Power Property for Exponents If a and b are real numbers and m is a whole number, then

To raise a product to a power, raise each factor to that power.

• Quotient to a Power Property for Exponents If $a$ and are real numbers, $b\ne{}0,$ and $m$ is an integer, then

To raise a fraction to a power, raise the numerator and denominator to that power.

• Summary of Exponent Properties If a and b are real numbers, and m and n are integers, then Property DescriptionProduct Property ${a}^{m}\cdot{a}^{n}={a}^{m+n}$ Power Property ${({a}^{m})}^{n}={a}^{m\cdot{}n}$ Product to a Power ${(ab)}^{n}={a}^{n}{b}^{n}$ Quotient Property $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},a\ne{}0$ Zero Exponent Property ${a}^{0}=1,a\ne{}0$ Quotient to a Power Property: ${(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne{}0$ Properties of Negative Exponents ${a}^{-n}=\frac{1}{{a}^{n}}$ and $\frac{1}{{a}^{-n}}={a}^{n}$ Quotient to a Negative Exponent ${(\frac{a}{b})}^{-n}={(\frac{b}{a})}^{n}$

• Scientific Notation A number is expressed in scientific notation when it is of the form

• How to convert a decimal to scientific notation. Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. Step 2. Count the number of decimal places, $n,$ that the decimal point was moved. Step 3. Write the number as a product with a power of 10. If the original number is. greater than 1, the power of 10 will be ${10}^{n}.$ between 0 and 1, the power of 10 will be ${10}^{-n}.$ Step 4. Check.

• How to convert scientific notation to decimal form. Step 1. Determine the exponent, $n,$ on the factor 10. Step 2. Move the decimal $n$ places, adding zeros if needed. If the exponent is positive, move the decimal point $n$ places to the right. If the exponent is negative, move the decimal point $|n|$ places to the left. Step 3. Check.

• How to use the FOIL method to multiply two binomials.

• Multiplying Two Binomials: To multiply binomials, use the: Distributive Property FOIL Method

• Multiplying a Polynomial by a Polynomial: To multiply a trinomial by a binomial, use the: Distributive Property Vertical Method

• Binomial Squares Pattern If a and b are real numbers,

• Product of Conjugates Pattern If $a,b$ are real numbers The product is called a difference of squares. To multiply conjugates, square the first term, square the last term, write it as a difference of squares.

• Comparing the Special Product Patterns Binomial Squares Product of Conjugates${(a+b)}^{2}={a}^{2}+2ab+{b}^{2}$$(a-b)(a+b)={a}^{2}-{b}^{2}$${(a-b)}^{2}={a}^{2}-2ab+{b}^{2}$ • Squaring a binomial • Multiplying conjugates • Product is a trinomial • Product is a binomial. • Inner and outer terms with FOIL are the same. • Inner and outer terms with FOIL are opposites. • Middle term is double the product of the terms • There is no middle term.

• Multiplication of Polynomial Functions: For functions $f(x)$ and $g(x),$

• Division of a Polynomial by a Monomial To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

• Division of Polynomial Functions For functions $f(x)$ and $g(x),$ where $g(x)\ne{}0,$$(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$

• Remainder Theorem If the polynomial function $f(x)$ is divided by $x-c,$ then the remainder is $f(c).$

• Factor Theorem: For any polynomial function $f(x),$ if $x-c$ is a factor of $f(x),$ then $f(c)=0$ if $f(c)=0,$ then $x-c$ is a factor of $f(x)$

##

Review Exercises

Types of Polynomials

In the following exercises, determine the type of polynomial.

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

In the following exercises, simplify.

Evaluate a Polynomial Function for a Given Value of the Variable

In the following exercises, find the function values for each polynomial function.

Add and Subtract Polynomial Functions

In the following exercises, find ⓐ (f + g)(x) ⓑ (f + g)(3) ⓒ (f − g)(x) ⓓ (f − g)(−2)

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

In the following exercises, simplify each expression using the Product Property.

In the following exercises, simplify each expression using the Power Property.

In the following exercises, simplify each expression using the Product to a Power Property.

In the following exercises, simplify each expression using the Quotient to a Power Property.

In the following exercises, simplify each expression by applying several properties.

In the following exercises, write each number in scientific notation.

In the following exercises, convert each number to decimal form.

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

Multiply Monomials

In the following exercises, multiply the monomials.

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using:

ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

In the following exercises, multiply the binomials. Use any method.

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

In the following exercises, multiply. Use either method.

Multiply Special Products

In the following exercises, square each binomial using the Binomial Squares Pattern.

In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

Divide Monomials

In the following exercises, divide the monomials.

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

Divide Polynomial Functions

In the following exercises, divide.

Use the Remainder and Factor Theorem

In the following exercises, use the Remainder Theorem to find the remainder.

In the following exercises, use the Factor Theorem to determine if $x-c$ is a factor of the polynomial function.

##

In the following exercises, simplify and write your answer in exponential notation.

Key Terms

binomial

A binomial is a polynomial with exactly two terms.

conjugate pair

degree of a constant

The degree of any constant is 0.

degree of a polynomial

The degree of a polynomial is the highest degree of all its terms.

degree of a term

The degree of a term is the sum of the exponents of its variables.

monomial

a

m

polynomial

A monomial or two or more monomials combined by addition or subtraction is a polynomial.

polynomial function

A polynomial function is a function whose range values are defined by a polynomial.

Power Property

a

m

n

a

m

n

Product Property

a

m

a

n

a

m

n

Product to a Power

a

b

m

a

m

b

m

Properties of Negative Exponents

a

n

a

n

a

n

a

n

Quotient Property

a

m

a

n

a

m

n

a

Quotient to a Negative Exponent

a

b

n

b

a

n

Quotient to a Power Property

a

b

m

a

m

b

m

b

standard form of a polynomial

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.

trinomial

A trinomial is a polynomial with exactly three terms.

Zero Exponent Property

a

a

##

Key Concepts

• Monomial A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form $a{x}^{m},$ where a is a constant and m is a whole number.

• Polynomials Polynomial—A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial. monomial —A polynomial with exactly one term is called a monomial. binomial — A polynomial with exactly two terms is called a binomial. trinomial —A polynomial with exactly three terms is called a trinomial.

• Degree of a Polynomial The degree of a term is the sum of the exponents of its variables. The degree of a constant is 0. The degree of a polynomial is the highest degree of all its terms.

• Exponential Notation This is read a to the ${m}^{th}$ power. In the expression ${a}^{m}$, the exponent m tells us how many times we use the base a as a factor.

• Product Property for Exponents If a is a real number and m and n are integers, then

To multiply with like bases, add the exponents.

• Quotient Property for Exponents If $a$ is a real number, $a\ne{}0,$ and m and n are integers, then

• Zero Exponent If a is a non-zero number, then ${a}^{0}=1.$ If a is a non-zero number, then a to the power of zero equals 1. Any non-zero number raised to the zero power is 1.

• Negative Exponent If n is an integer and $a\ne{}0,$ then ${a}^{-n}=\frac{1}{{a}^{n}}$ or $\frac{1}{{a}^{-n}}={a}^{n}.$

• Quotient to a Negative Exponent Property If $a,b$ are real numbers, $a\ne{}0,b\ne{}0$ and $n$ is an integer, then

• Power Property for Exponents If $a$ is a real number and $m,n$ are integers, then

To raise a power to a power, multiply the exponents.

• Product to a Power Property for Exponents If a and b are real numbers and m is a whole number, then

To raise a product to a power, raise each factor to that power.

• Quotient to a Power Property for Exponents If $a$ and are real numbers, $b\ne{}0,$ and $m$ is an integer, then

To raise a fraction to a power, raise the numerator and denominator to that power.

• Summary of Exponent Properties If a and b are real numbers, and m and n are integers, then Property DescriptionProduct Property ${a}^{m}\cdot{a}^{n}={a}^{m+n}$ Power Property ${({a}^{m})}^{n}={a}^{m\cdot{}n}$ Product to a Power ${(ab)}^{n}={a}^{n}{b}^{n}$ Quotient Property $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},a\ne{}0$ Zero Exponent Property ${a}^{0}=1,a\ne{}0$ Quotient to a Power Property: ${(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne{}0$ Properties of Negative Exponents ${a}^{-n}=\frac{1}{{a}^{n}}$ and $\frac{1}{{a}^{-n}}={a}^{n}$ Quotient to a Negative Exponent ${(\frac{a}{b})}^{-n}={(\frac{b}{a})}^{n}$

• Scientific Notation A number is expressed in scientific notation when it is of the form

• How to convert a decimal to scientific notation. Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. Step 2. Count the number of decimal places, $n,$ that the decimal point was moved. Step 3. Write the number as a product with a power of 10. If the original number is. greater than 1, the power of 10 will be ${10}^{n}.$ between 0 and 1, the power of 10 will be ${10}^{-n}.$ Step 4. Check.

• How to convert scientific notation to decimal form. Step 1. Determine the exponent, $n,$ on the factor 10. Step 2. Move the decimal $n$ places, adding zeros if needed. If the exponent is positive, move the decimal point $n$ places to the right. If the exponent is negative, move the decimal point $|n|$ places to the left. Step 3. Check.

• How to use the FOIL method to multiply two binomials.

• Multiplying Two Binomials: To multiply binomials, use the: Distributive Property FOIL Method

• Multiplying a Polynomial by a Polynomial: To multiply a trinomial by a binomial, use the: Distributive Property Vertical Method

• Binomial Squares Pattern If a and b are real numbers,

• Product of Conjugates Pattern If $a,b$ are real numbers The product is called a difference of squares. To multiply conjugates, square the first term, square the last term, write it as a difference of squares.

• Comparing the Special Product Patterns Binomial Squares Product of Conjugates${(a+b)}^{2}={a}^{2}+2ab+{b}^{2}$$(a-b)(a+b)={a}^{2}-{b}^{2}$${(a-b)}^{2}={a}^{2}-2ab+{b}^{2}$ • Squaring a binomial • Multiplying conjugates • Product is a trinomial • Product is a binomial. • Inner and outer terms with FOIL are the same. • Inner and outer terms with FOIL are opposites. • Middle term is double the product of the terms • There is no middle term.

• Multiplication of Polynomial Functions: For functions $f(x)$ and $g(x),$

• Division of a Polynomial by a Monomial To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

• Division of Polynomial Functions For functions $f(x)$ and $g(x),$ where $g(x)\ne{}0,$$(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$

• Remainder Theorem If the polynomial function $f(x)$ is divided by $x-c,$ then the remainder is $f(c).$

• Factor Theorem: For any polynomial function $f(x),$ if $x-c$ is a factor of $f(x),$ then $f(c)=0$ if $f(c)=0,$ then $x-c$ is a factor of $f(x)$

##

Review Exercises

Types of Polynomials

In the following exercises, determine the type of polynomial.

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

In the following exercises, simplify.

Evaluate a Polynomial Function for a Given Value of the Variable

In the following exercises, find the function values for each polynomial function.

Add and Subtract Polynomial Functions

In the following exercises, find ⓐ (f + g)(x) ⓑ (f + g)(3) ⓒ (f − g)(x) ⓓ (f − g)(−2)

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

In the following exercises, simplify each expression using the Product Property.

In the following exercises, simplify each expression using the Power Property.

In the following exercises, simplify each expression using the Product to a Power Property.

In the following exercises, simplify each expression using the Quotient to a Power Property.

In the following exercises, simplify each expression by applying several properties.

In the following exercises, write each number in scientific notation.

In the following exercises, convert each number to decimal form.

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

Multiply Monomials

In the following exercises, multiply the monomials.

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using:

ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

In the following exercises, multiply the binomials. Use any method.

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

In the following exercises, multiply. Use either method.

Multiply Special Products

In the following exercises, square each binomial using the Binomial Squares Pattern.

In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

Divide Monomials

In the following exercises, divide the monomials.

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

Divide Polynomial Functions

In the following exercises, divide.

Use the Remainder and Factor Theorem

In the following exercises, use the Remainder Theorem to find the remainder.

In the following exercises, use the Factor Theorem to determine if $x-c$ is a factor of the polynomial function.

##

In the following exercises, simplify and write your answer in exponential notation.

1. $5m+9$

2. ${y}^{2}+6{y}^{3}+9{y}^{4}$

3. $-8{y}^{3}-5{y}^{3}$

4. $(8{m}^{2}+12m-5)-(2{m}^{2}-7m-1)$

5. $(5{u}^{2}+8u)-(4u-7)$

6. Find the difference of ${x}^{2}+6x+8$ and ${x}^{2}-8x+15.$

1. $18a-7b-21a$

2. $(6{a}^{2}+7)+(2{a}^{2}-5a-9)$

3. $(7{m}^{2}-2m-5)-(4{m}^{2}+m-8)$

4. Subtract $(8{y}^{2}-y+9)$ from $(11{y}^{2}-9y-5)$

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

6. For the function $f(x)=7{x}^{2}-3x+5$ find:ⓐ $f(5)$ ⓑ $f(-2)$ ⓒ $f(0)$

1. A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function $h(t)=-16{t}^{2}+640$ gives the height of the glasses t seconds after they were dropped. Find the height of the glasses when $t=6.$

2. $f(x)=2{x}^{2}-4x-7$ and $g(x)=2{x}^{2}-x+5$

3. ${p}^{3}\cdot{p}^{10}$

4. $a\cdot{a}^{2}\cdot{a}^{3}$

5. ${y}^{a}\cdot{y}^{b}$

6. $\frac{{a}^{6}}{a}$

1. $\frac{1}{{x}^{5}}$

2. $12{a}^{0}-15{b}^{0}$

3. ${(-10)}^{-3}$

4. ${(8n)}^{-1}$

5. ${10}^{-3}$

6. ${(-\frac{1}{5})}^{-3}$

1. ${(-5)}^{-3}$

2. ${(-\frac{3}{x})}^{-3}$

3. ${({3}^{2})}^{5}$

4. ${x}^{-3}\cdot{x}^{9}$

5. $(u{v}^{-3})({u}^{-4}{v}^{-2})$

6. ${p}^{5}\cdot{p}^{-2}\cdot{p}^{-4}$

1. $\frac{{q}^{4}}{{q}^{20}}$

2. $\frac{{n}^{-3}}{{n}^{-5}}$

3. ${(3{y}^{-4})}^{2}$

4. ${(\frac{3x{y}^{2}}{z})}^{4}$

5. ${({x}^{2}y)}^{2}{(3x{y}^{5})}^{3}$

6. ${(\frac{3x{y}^{3}}{4{x}^{4}{y}^{-2}})}^{2}{(\frac{6x{y}^{4}}{8{x}^{3}{y}^{-2}})}^{-1}$

1. 5,300,000

2. $2.9\,\times\,{10}^{4}$

3. $9.413\,\times\,{10}^{-5}$

4. $(1.5\,\times\,{10}^{-3})(4.8\,\times\,{10}^{-1})$

5. $\frac{9\,\times\,{10}^{-3}}{1\,\times\,{10}^{-6}}$

6. $(\frac{1}{3}{c}^{2})(30{c}^{8})$

1. $(\frac{2}{3}{m}^{3}{n}^{6})(\frac{1}{6}{m}^{4}{n}^{4})$

2. ${a}^{2}({a}^{2}-9a-36)$

3. $(4n-5)(2{n}^{3})$

4. $(y-4)(y+12)$

5. $(6p-11)(3p-10)$

6. $(k+6)(k-9)$

1. $(2y-9)(5y-7)$

2. $(x-8)(x+9)$

3. $(10a-1)(3a-3)$

4. $(5b-2)(3{b}^{2}+b-9)$

5. $(4y-1)(6{y}^{2}-12y+5)$

6. ${(x+\frac{3}{4})}^{2}$

1. ${(5p+7q)}^{2}$

2. $(6x+y)(6x-y)$

3. $(12{x}^{3}-7{y}^{2})(12{x}^{3}+7{y}^{2})$

4. $72{p}^{12}\mathrm{÷}8{p}^{3}$

5. $\frac{45{y}^{6}}{-15{y}^{10}}$

6. $\frac{28{a}^{9}b}{7{a}^{4}{b}^{3}}$

1. $\frac{(5{m}^{9}{n}^{3})(8{m}^{3}{n}^{2})}{(10m{n}^{4})({m}^{2}{n}^{5})}$

2. $(54{y}^{4}-24{y}^{3})\mathrm{÷}(-6{y}^{2})$

3. $\frac{12{x}^{2}+4x-3}{-4x}$

4. $({y}^{2}+2y+18)\mathrm{÷}(y+5)$

5. $({a}^{3}-1)\mathrm{÷}(a+1)$

6. $2{x}^{3}-11{x}^{2}+11x+12$ is divided by $x-3$

1. For functions $f(x)={x}^{2}-15x+54$ and $g(x)=x-9,$ find ⓐ $(\frac{f}{g})(x)$ⓑ $(\frac{f}{g})(-2)$

2. $f(x)={x}^{3}-4x-9$ is divided by $x+2$

3. Determine whether $x-2$ is a factor of ${x}^{3}-7{x}^{2}+7x-6$.

4. For the polynomial $8{y}^{4}-3{y}^{2}+1$

ⓐ Is it a monomial, binomial, or trinomial? ⓑ What is its degree?

5. $(10{x}^{2}-3x+5)-(4{x}^{2}-6)$

6. ${x}^{-3}{x}^{4}$

1. ${(47{a}^{18}{b}^{23}{c}^{5})}^{0}$

2. ${(2y)}^{-3}$

3. $\frac{{x}^{4}}{{x}^{-5}}$

4. $\frac{24{r}^{3}s}{6{r}^{2}{s}^{7}}$

5. $(8x{y}^{3})(-6{x}^{4}{y}^{6})$

6. $(m+3)(7m-2)$

1. ${(4x-3)}^{2}$

2. $(15x{y}^{3}-35{x}^{2}y)\mathrm{÷}5xy$

3. Use the Factor Theorem to determine if $x+3$ a factor of ${x}^{3}+8{x}^{2}+21x+18.$

4. $(2.4\,\times\,{10}^{8})(2\,\times\,{10}^{-5})$

5. For the function $f(x)=6{x}^{2}-3x-9$ find:ⓐ $f(3)$ ⓑ $f(-2)$ ⓒ $f(0)$

6. For functions$f(x)=3{x}^{2}-23x-36$ and$g(x)=x-9,$ findⓐ $(\frac{f}{g})(x)$ ⓑ $(\frac{f}{g})(3)$