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

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 exponent of its variable.

monomial

negative exponent

polynomial

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

scientific notation

standard form

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.

##

Key Concepts

• Monomials A monomial 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 monomials 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

• Properties of Exponents If $a,b$ are real numbers and $m,n$ are whole numbers, then$\begin{array}{lccrll} \mathrm{Product Property} & & & {a}^{m}\cdot{a}^{n} & = & {a}^{m+n} \\ \mathrm{Power Property} & & & {({a}^{m})}^{n} & = & {a}^{m\cdot{}n} \\ \mathrm{Product to a Power} & & & {(ab)}^{m} & = & {a}^{m}{b}^{m} \\ \end{array}$

• FOIL Method for Multiplying Two Binomials—To multiply two binomials: Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last terms.

• Multiplying Two Binomials—To multiply binomials, use the: Distributive Property (Example 6.34) FOIL Method (Example 6.39) Vertical Method (Example 6.44)

• Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the: Distributive Property (Example 6.45) Vertical Method (Example 6.46)

• Binomial Squares Pattern If $a,b$ are real numbers, ${(a+b)}^{2}={a}^{2}+2ab+{b}^{2}$${(a-b)}^{2}={a}^{2}-2ab+{b}^{2}$ To square a binomial: square the first term, square the last term, double their product.

• Product of Conjugates Pattern If $a,b$ are real numbers, $(a-b)(a+b)={a}^{2}-{b}^{2}$ 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

• Quotient Property for Exponents: If $a$ is a real number, $a\ne{}0$, and $m,n$ are whole numbers, then:$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\,\mathrm{and}\,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{m-n}},n>m$

• Zero Exponent If $a$ is a non-zero number, then ${a}^{0}=1$.

• Quotient to a Power Property for Exponents: If $a$ and $b$ are real numbers, $b\ne{}0,$ and $m$ is a counting number, then:${(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}$ To raise a fraction to a power, raise the numerator and denominator to that power.

• Summary of Exponent Properties If $a,b$ are real numbers and $m,n$ are whole numbers, then$\begin{array}{lcrll} \mathrm{Product Property} & & {a}^{m}\cdot{a}^{n} & = & {a}^{m+n} \\ \mathrm{Power Property} & & {({a}^{m})}^{n} & = & {a}^{m\cdot{}n} \\ \mathrm{Product to a Power} & & {(ab)}^{m} & = & {a}^{m}{b}^{m} \\ \mathrm{Quotient Property} & & \frac{{a}^{m}}{{a}^{n}} & = & {a}^{m-n},a\ne{}0,m>n \\ & & \frac{{a}^{m}}{{a}^{n}} & = & \frac{1}{{a}^{n-m}},a\ne{}0,n>m \\ \mathrm{Zero Exponent Definition} & & {a}^{o} & = & 1,a\ne{}0 \\ \mathrm{Quotient to a Power Property} & & {(\frac{a}{b})}^{m} & = & \frac{{a}^{m}}{{b}^{m}},b\ne{}0 \\ \end{array}$

• Fraction Addition If $a,b,\,\mathrm{and}\,c$ are numbers where $c\ne{}0$, then$\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\,\mathrm{and}\,\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$

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

• Property of Negative Exponents If $n$ is a positive integer and $a\ne{}0$, then $\frac{1}{{a}^{-n}}={a}^{n}$

• Quotient to a Negative Exponent If $a,b$ are real numbers, $b\ne{}0$ and $n$ is an integer , then ${(\frac{a}{b})}^{-n}={(\frac{b}{a})}^{n}$

• 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.

• 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.

##

Review Exercises

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

Simplify Expressions with Exponents

In the following exercises, simplify.

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

Multiply Monomials

In the following exercises 8, 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 Trinomial by a Binomial

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

In the following exercises, multiply. Use either method.

Square a Binomial Using the Binomial Squares Pattern

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

Multiply Conjugates Using the Product of Conjugates Pattern

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

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

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 a Polynomial by a Binomial

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

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Convert from Decimal Notation to Scientific Notation

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

Convert Scientific Notation to Decimal Form

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

Multiply and Divide Using Scientific Notation

In the following exercises, multiply and write your answer in decimal form.

In the following exercises, divide and write your answer in decimal form.

##

In the following exercises, simplify each expression.

In the following exercises, simplify, and write your answer in decimal form.

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 exponent of its variable.

monomial

negative exponent

polynomial

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

scientific notation

standard form

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.

##

Key Concepts

• Monomials A monomial 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 monomials 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

• Properties of Exponents If $a,b$ are real numbers and $m,n$ are whole numbers, then$\begin{array}{lccrll} \mathrm{Product Property} & & & {a}^{m}\cdot{a}^{n} & = & {a}^{m+n} \\ \mathrm{Power Property} & & & {({a}^{m})}^{n} & = & {a}^{m\cdot{}n} \\ \mathrm{Product to a Power} & & & {(ab)}^{m} & = & {a}^{m}{b}^{m} \\ \end{array}$

• FOIL Method for Multiplying Two Binomials—To multiply two binomials: Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last terms.

• Multiplying Two Binomials—To multiply binomials, use the: Distributive Property (Example 6.34) FOIL Method (Example 6.39) Vertical Method (Example 6.44)

• Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the: Distributive Property (Example 6.45) Vertical Method (Example 6.46)

• Binomial Squares Pattern If $a,b$ are real numbers, ${(a+b)}^{2}={a}^{2}+2ab+{b}^{2}$${(a-b)}^{2}={a}^{2}-2ab+{b}^{2}$ To square a binomial: square the first term, square the last term, double their product.

• Product of Conjugates Pattern If $a,b$ are real numbers, $(a-b)(a+b)={a}^{2}-{b}^{2}$ 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

• Quotient Property for Exponents: If $a$ is a real number, $a\ne{}0$, and $m,n$ are whole numbers, then:$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\,\mathrm{and}\,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{m-n}},n>m$

• Zero Exponent If $a$ is a non-zero number, then ${a}^{0}=1$.

• Quotient to a Power Property for Exponents: If $a$ and $b$ are real numbers, $b\ne{}0,$ and $m$ is a counting number, then:${(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}$ To raise a fraction to a power, raise the numerator and denominator to that power.

• Summary of Exponent Properties If $a,b$ are real numbers and $m,n$ are whole numbers, then$\begin{array}{lcrll} \mathrm{Product Property} & & {a}^{m}\cdot{a}^{n} & = & {a}^{m+n} \\ \mathrm{Power Property} & & {({a}^{m})}^{n} & = & {a}^{m\cdot{}n} \\ \mathrm{Product to a Power} & & {(ab)}^{m} & = & {a}^{m}{b}^{m} \\ \mathrm{Quotient Property} & & \frac{{a}^{m}}{{a}^{n}} & = & {a}^{m-n},a\ne{}0,m>n \\ & & \frac{{a}^{m}}{{a}^{n}} & = & \frac{1}{{a}^{n-m}},a\ne{}0,n>m \\ \mathrm{Zero Exponent Definition} & & {a}^{o} & = & 1,a\ne{}0 \\ \mathrm{Quotient to a Power Property} & & {(\frac{a}{b})}^{m} & = & \frac{{a}^{m}}{{b}^{m}},b\ne{}0 \\ \end{array}$

• Fraction Addition If $a,b,\,\mathrm{and}\,c$ are numbers where $c\ne{}0$, then$\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\,\mathrm{and}\,\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$

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

• Property of Negative Exponents If $n$ is a positive integer and $a\ne{}0$, then $\frac{1}{{a}^{-n}}={a}^{n}$

• Quotient to a Negative Exponent If $a,b$ are real numbers, $b\ne{}0$ and $n$ is an integer , then ${(\frac{a}{b})}^{-n}={(\frac{b}{a})}^{n}$

• 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.

• 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.

##

Review Exercises

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

Simplify Expressions with Exponents

In the following exercises, simplify.

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

Multiply Monomials

In the following exercises 8, 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 Trinomial by a Binomial

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

In the following exercises, multiply. Use either method.

Square a Binomial Using the Binomial Squares Pattern

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

Multiply Conjugates Using the Product of Conjugates Pattern

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

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

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 a Polynomial by a Binomial

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

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Convert from Decimal Notation to Scientific Notation

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

Convert Scientific Notation to Decimal Form

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

Multiply and Divide Using Scientific Notation

In the following exercises, multiply and write your answer in decimal form.

In the following exercises, divide and write your answer in decimal form.

##

In the following exercises, simplify each expression.

In the following exercises, simplify, and write your answer in decimal form.

1. ⓐ ${a}^{2}-{b}^{2}$ ⓑ $24{d}^{3}$ ⓒ ${x}^{2}+8x-10$ ⓓ ${m}^{2}{n}^{2}-2mn+6$ ⓔ $7{y}^{3}+{y}^{2}-2y-4$

2. 1. ⓐ $5{p}^{3}-8{p}^{2}+10p-4$

2. ⓑ $-20{q}^{4}$

3. ⓒ ${x}^{2}+6x+12$

4. ⓓ $23{r}^{2}{s}^{2}-4rs+5$

5. ⓔ 100

3. $-14k+19k$

4. $-9c-18c$

5. $3{m}^{2}+7{n}^{2}-3{m}^{2}$

6. $\mathrm{13a}+b$

1. $(9{p}^{2}-5p+3)+(4{p}^{2}-4)$

2. $(7{y}^{2}-8y)-(y-4)$

3. Find the sum of $({a}^{2}+6a+9)\,\mathrm{and}\,(5{a}^{3}-7)$

4. Evaluate $10-12x$ when:

1. ⓐ $x=3$

2. ⓑ $x=0$

3. ⓒ $x=-1$

5. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $-4{p}^{2}+460p.$ Find the revenue received when $p=75$ dollars.

6. $-{2}^{6}$

1. ${p}^{15}\cdot{p}^{16}$

2. $8\cdot{8}^{5}$

3. ${y}^{c}\cdot{y}^{3}$

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

5. ${({3}^{r})}^{s}$

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

1. ${(10xyz)}^{3}$

2. ${(4{a}^{3}{b}^{2})}^{3}$

3. ${(2{q}^{3})}^{4}{(3q)}^{2}$

4. ${(\frac{2}{5}{m}^{2}n)}^{3}$

5. $(-9{n}^{7})(-16n)$

6. $(\frac{5}{9}a{b}^{2})(27a{b}^{3})$

1. $-4(y+13)$

2. $p(p+3)$

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

4. $3{q}^{2}({q}^{2}-7q+6)$ 3

5. $(b-4)\cdot{}11$

6. $(6y-7)(2y-5)$

1. $(y-4)(y-8)$

2. $(q+16)(q-3)$

3. $({u}^{2}+6)({u}^{2}-5)$

4. $(8mn+3)(2mn-1)$

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

6. $(7m+1)({m}^{2}-10m-3)$

1. ${(q-15)}^{2}$

2. ${(8u+1)}^{2}$

3. ${(4a-3b)}^{2}$

4. $(s-7)(s+7)$

5. $(y+\frac{2}{5})(y-\frac{2}{5})$

6. $(6-r)(6+r)$

1. $(5{p}^{4}-4{q}^{3})(5{p}^{4}+4{q}^{3})$

2. $(6a+11)(6a-11)$

3. ${({c}^{4}+9d)}^{2}$

4. $({a}^{2}+4b)(4a-{b}^{2})$

5. $\frac{{10}^{25}}{{10}^{5}}$

6. $\frac{{v}^{12}}{{v}^{48}}$

1. $\frac{5}{{5}^{8}}$

2. $(-{12}^{0})$${(-12)}^{0}$

3. ${(\frac{m}{3})}^{4}$

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

5. ${(\frac{{r}^{8}}{{r}^{3}})}^{4}$

6. ${(\frac{3{x}^{4}}{2{y}^{2}})}^{5}$

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

2. $\frac{64{a}^{5}{b}^{9}}{-16{a}^{10}{b}^{3}}$

3. $\frac{(8{p}^{6}{q}^{2})(9{p}^{3}{q}^{5})}{16{p}^{8}{q}^{7}}$

4. $(35{x}^{2}-75x)\mathrm{÷}5x$

5. $\frac{550{p}^{6}-300{p}^{4}}{10{p}^{3}}$

6. $\frac{96{a}^{5}{b}^{2}-48{a}^{4}{b}^{3}-56{a}^{2}{b}^{4}}{8a{b}^{2}}$

1. $\frac{105{y}^{5}+50{y}^{3}-5y}{5{y}^{3}}$

2. $({v}^{2}-16v+64)\mathrm{÷}(v-8)$

3. $({n}^{2}-3n-14)\mathrm{÷}(n+3)$

4. $({u}^{3}-8)\mathrm{÷}(u-2)$

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

6. ${(6u)}^{-3}$

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

2. ${q}^{-6}\cdot{q}^{-5}$

3. ${({y}^{8})}^{-1}$

4. $\frac{{a}^{8}}{{a}^{12}}$

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

6. In 2015, the population of the world was about 7,200,000,000 people.

1. $1.5\,\times\,{10}^{10}$

2. $5.5\,\times\,{10}^{-1}$

3. $(3.5\,\times\,{10}^{-2})(6.2\,\times\,{10}^{-1})$

4. $\frac{9\,\times\,{10}^{-5}}{3\,\times\,{10}^{2}}$

5. $(12{a}^{2}-7a+4)+(3{a}^{2}+8a-10)$

6. ${(-\frac{2}{5})}^{3}$

1. ${(4{a}^{3}{b}^{5})}^{2}$

2. $3k({k}^{2}-7k+13)$

3. $(v-9)(9v-5)$

4. $(n-6)({n}^{2}-5n+4)$

5. $(7p-5)(7p+5)$

6. $\frac{{3}^{8}}{{3}^{10}}$

1. ${(87{x}^{15}{y}^{3}{z}^{22})}^{0}$

2. $\frac{12{x}^{2}+42x-6}{2x}$

3. $\frac{64{x}^{3}-1}{4x-1}$

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

5. ${q}^{-4}\cdot{q}^{-5}$

6. Convert 83,000,000 to scientific notation.

1. $(3.4\,\times\,{10}^{9})(2.2\,\times\,{10}^{-5})$

2. A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial $-16{t}^{2}+1000$ gives the height of the package $t$ seconds a after it was dropped. Find the height when $t=6$ seconds.