CHAPTER OUTLINE

10Exponents and Polynomials

Slide 2Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

10.1 Addition and Subtraction of Polynomials10.2 Multiplication Properties of Exponents10.3 Multiplication of Polynomials10.4 Introduction to Factoring10.5 Negative Exponents and the Quotient Rule for

Exponents10.6 Scientific Notation

Section

Objectives

10.1 Addition and Subtraction of Polynomials

Slide 3Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

1. Key Definitions2. Addition of Polynomials3. Subtraction of Polynomials4. Evaluating Polynomials and Applications

Section 10.1 Addition and Subtraction of Polynomials

1. Key Definitions

Slide 4Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Recall that a term is a number or a product or quotient of numbers and variables.

A term in which the variables appear only in the numerator with whole number exponents is called a monomial.

A polynomial is one or more monomials combined by addition or subtraction.

DEFINITION Categorizing Polynomials

Slide 5Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

• If a polynomial has exactly one term, then it is called a monomial.Example: 3xy4 (1 term)

• If a polynomial has exactly two terms, then it is called a binomial.Example: 5ab + 6 (2 terms)

• If a polynomial has exactly three terms, then it is called a trinomial.Example: 6x4 – 7x2 – 5x (3 terms)

Section 10.1 Addition and Subtraction of Polynomials

1. Key Definitions

Slide 6Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

This is written in descending order.

The degree is 6.

Example 1 Identifying the Characteristics of a Polynomial

Slide 7Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Write each polynomial in descending order. Determine the degree of the polynomial, and categorize the polynomial as a monomial, binomial, ortrinomial.

ExampleSolution:

1 Identifying the Characteristics of a Polynomial

Slide 8Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 10.1 Addition and Subtraction of Polynomials

1. Key Definitions

Slide 9Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The coefficient of a term is the numerical factorof the term.

Section 10.1 Addition and Subtraction of Polynomials

2. Addition of Polynomials

Slide 10Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Like terms have the same variables, raised to the same powers.

Example 2 Combining Like Terms

Slide 11Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

2 Combining Like Terms

Slide 12Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 10.1 Addition and Subtraction of Polynomials

2. Addition of Polynomials

Slide 13Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

To add two polynomials, first use the associative and commutative properties of addition to group like terms. Then combine like terms.

Example 3 Adding Polynomials

Slide 14Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

3 Adding Polynomials

Slide 15Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

TIP:

Slide 16Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Polynomials can also be added vertically. Begin by lining up like terms in the same column.

Section 10.1 Addition and Subtraction of Polynomials

3. Subtraction of Polynomials

Slide 17Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Recall that subtraction of real numbers is defined as a – b = a + (–b). That is, we add the opposite of the second number to the first number. We will use the same strategy to subtract polynomials. To find the opposite of a polynomial, take the opposite of each term.

Example 5 Finding the Opposite of a Polynomial

Slide 18Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Write the opposite of the polynomial.

ExampleSolution:

5 Finding the Opposite of a Polynomial

Slide 19Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The opposite of a real number a is written as (–a). We can apply the same notation to find the opposite of a polynomial.

The opposite of 2x2 – 7x + 5 is –(2x2 – 7x + 5).

= –2x2 + 7x – 5 Apply the distributive property.

Example 6 Subtracting Polynomials

Slide 20Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

6 Subtracting Polynomials

Slide 21Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Write the opposite of the second polynomial by applying the distributive property.

Regroup and collect like terms.

Combine like terms.

TIP:

Slide 22Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Subtraction of polynomials can also be performed vertically. To do so, add the opposite of the second polynomial to the first polynomial.

Example 7 Subtracting Polynomials

Slide 23Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

7 Subtracting Polynomials

Slide 24Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 10.1 Addition and Subtraction of Polynomials

4. Evaluating Polynomials and Applications

Slide 25Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

A polynomial is an algebraic expression. Evaluating a polynomial for a value of the variable is the same as evaluating an expression.

Example 8 Evaluating a Polynomial

Slide 26Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Evaluate the polynomial for the given value of the variable:

ExampleSolution:

8 Evaluating a Polynomial

Slide 27Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 9 Evaluating a Polynomial in an Application

Slide 28Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The cost (in dollars) to rent storage space for x months is given by the polynomial.

49.99x + 129

a. Evaluate the polynomial for x = 3 and interpret the result in the context of the problem.b. Evaluate the polynomial for x = 12 and interpret the result in the context of the problem.

ExampleSolution:

9 Evaluating a Polynomial in an Application

Slide 29Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section

Objectives

10.2 Multiplication Properties of Exponents

Slide 30Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

1. Multiplication of Like Bases: am an = amn

2. Multiplying Monomials3. Power Rule of Exponents: (am)n = am n

4. The Power of a Product and the Power of a Quotient

PROPERTY Multiplication of Factors with Like Bases

Slide 31Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 1 Multiplying Factors with the Same Base

Slide 32Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

1 Multiplying Factors with the Same Base

Slide 33Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The base remains unchanged. Add the exponents.

Section 10.2 Multiplication Properties of Exponents

2. Multiplying Monomials

Slide 34Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

We use the commutative and associative properties of multiplication to regroup factors and multiply like bases.

Example 2 Multiplying Monomials

Slide 35Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

2 Multiplying Monomials

Slide 36Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 3 Multiplying Monomials

Slide 37Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

3 Multiplying Monomials

Slide 38Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

PROPERTY Power Rule for Exponents

Slide 39Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 4 Applying the Power Rule for Exponents

Slide 40Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Simplify the expressions.

ExampleSolution:

4 Applying the Power Rule for Exponents

Slide 41Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Avoiding Mistakes

Slide 42Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

PROPERTY Power of a Product and the Power of a Quotient

Slide 43Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 5 Simplifying a Power of a Product or Quotient

Slide 44Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Simplify.

ExampleSolution:

5 Simplifying a Power of a Product or Quotient

Slide 45Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 6 Simplifying Expressions Involving Exponents

Slide 46Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

6 Simplifying Expressions Involving Exponents

Slide 47Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Apply the power rule to each factor in parentheses. Operations with exponents are performed before multiplication.

Section

Objectives

10.3 Multiplication of Polynomials

Slide 48Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

1. Multiplying a Monomial by a Polynomial2. Multiplying a Polynomial by a Polynomial

Section 10.3 Multiplication of Polynomials

1. Multiplying a Monomial by a Polynomial

Slide 49Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

To multiply a monomial by a polynomial with more than one term, use the distributive property.

Example 1 Multiplying a Monomial by a Polynomial

Slide 50Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

1 Multiplying a Monomial by a Polynomial

Slide 51Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 2 Multiplying a Monomial by a Polynomial

Slide 52Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

2 Multiplying a Monomial by a Polynomial

Slide 53Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

PROCEDURE Multiplying Polynomials

Slide 54Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Step 1 To multiply two polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

Step 2 Combine like terms if possible.

Example 4 Multiplying a Binomial by a Binomial

Slide 55Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

4 Multiplying a Binomial by a Binomial

Slide 56Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Multiply each term in the first binomial by each term in the second binomial.

TIP:

Slide 57Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Notice that the product of two binomials equals the sum of the products of the First, Outer, Inner, and Last terms. The word “FOIL” can be used as a memory device to multiply two binomials.

Note that FOIL only works when multiplyingbinomials.

Example 5 Multiplying a Binomial by a Binomial

Slide 58Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

5 Multiplying a Binomial by a Binomial

Slide 59Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Multiply each term in the first binomial by each term in thesecond binomial.

Simplify each term. Notice that there are no like terms to combine.

Example 7 Squaring a Binomial

Slide 60Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

7 Squaring a Binomial

Slide 61Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

To square a quantity, multiplythe quantity times itself.

Example 8 Multiplying a Polynomial by a Polynomial

Slide 62Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

8 Multiplying a Polynomial by a Polynomial

Slide 63Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Apply the distributive property. Multiply each term in the first polynomial by each term in the second.

Section

Objectives

10.4 Introduction to Factoring

Slide 64Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

1. Greatest Common Factor2. Factoring Out the Greatest Common Factor

Section 10.4 Introduction to Factoring

1. Greatest Common Factor

Slide 65Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The greatest common factor (denoted GCF) of two or more integers is the greatest factor that divides evenly into each integer.

PROCEDURE Finding the GCF of Two or More Integers

Slide 66Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Step 1 Factor each integer into prime factors.Step 2 Determine the prime factors common to each integer, including repeated factors.Step 3 The product of the factors from step 2 is the GCF

Example 1 Determining the Greatest Common Factor

Slide 67Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Determine the GCF.

ExampleSolution:

1 Determining the Greatest Common Factor

(continued)

Slide 68Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

1 Determining the Greatest Common Factor

Slide 69Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 2 Determining the Greatest Common Factor

Slide 70Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Determine the GCF.

ExampleSolution:

2 Determining the Greatest Common Factor

(continued)

Slide 71Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

2 Determining the Greatest Common Factor

Slide 72Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 3 Factoring Out the GCF

Slide 73Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

3 Factoring Out the GCF

Slide 74Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 5 Factoring Out the GCF

Slide 75Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

5 Factoring Out the GCF

Slide 76Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section

Objectives

10.5 Negative Exponents and the Quotient Rule for Exponents

Slide 77Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

1. Division of Like Bases2. Definition of b0

3. Definition of b–n

4. Properties of Exponents: A Summary

PROPERTY Division of Like Bases

Slide 78Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 1 Dividing Like Bases

Slide 79Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

1 Dividing Like Bases

Slide 80Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The base is unchanged. Subtract the exponents.

DEFINITION b0

Slide 81Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Let b be a nonzero number. Then b0 = 1.

Example 2 Simplifying Expressions with a Zero Exponent

Slide 82Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

2 Simplifying Expressions with a Zero Exponent

Slide 83Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

DEFINITION b –n

Slide 84Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 10.5 Negative Exponents and the Quotient Rule for Exponents

3. Definition of b–n

Slide 85Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

To evaluate b–n, take the reciprocal of the base and change the sign of the exponent.

Example 3 Simplifying Expressions Containing Negative Exponents

Slide 86Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

3 Simplifying Expressions Containing Negative Exponents

(continued)

Slide 87Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Take the reciprocal of the base. Change the sign of the exponent.

ExampleSolution:

3 Simplifying Expressions Containing Negative Exponents

Slide 88Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 4 Simplifying Expressions Containing Negative Exponents

Slide 89Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

4 Simplifying Expressions Containing Negative Exponents

(continued)

Slide 90Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Take the reciprocal of the base. Change the sign of the exponent.

Square the numerator and square the denominator.

ExampleSolution:

4 Simplifying Expressions Containing Negative Exponents

Slide 91Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

There are no parentheses to group the 5 and x as a single base. Therefore, the exponent of –2 applies only to x. The factor of 5 has an implied exponent of 1.

Example 5 Simplifying Expressions ContainingNegative Exponents

Slide 92Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

5 Simplifying Expressions ContainingNegative Exponents

Slide 93Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The negative exponent on y changes the position of y within the fraction. Notice that x does not change position because its exponent is positive.

The factor of a and the factor of c have negative exponents. Change their positions within the fraction. Notice that b does not change position because its exponent is positive.

Section 10.5 Negative Exponents and the Quotient Rule for Exponents

4. Properties of Exponents: A Summary

Slide 94Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 6 Simplifying Expressions Containing Exponents

Slide 95Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ExampleSolution:

6 Simplifying Expressions Containing Exponents

(continued)

Slide 96Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The base is unchanged. Add the exponents (property 1).

The base is unchanged. Subtract the exponents (property 2).

ExampleSolution:

6 Simplifying Expressions Containing Exponents

Slide 97Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section

Objectives

10.6 Scientific Notation

Slide 98Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

1. Scientific Notation2. Converting to Scientific Notation3. Converting Scientific Notation to Standard Form

Section 10.6 Scientific Notation

1. Scientific Notation

Slide 99Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Scientific notation is a means by which we can write very large numbers and very small numbers without having to write numerous zeros in the number.

Section 10.6 Scientific Notation

1. Scientific Notation

Slide 100Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Using scientific notation we express a number as the product of two factors. One factor is a number greater than or equal to 1, but less than 10. The other factor is a power of 10.

DEFINITION Scientific Notation

Slide 101Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

A positive number written in scientific notation is written as a 10n, where a is a number greater than or equal to 1, but less than 10, and n is an integer.

Section 10.6 Scientific Notation

2. Converting to Scientific Notation

(continued)

Slide 102Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

To write a number in scientific notation, follow these guidelines.

Move the decimal point so that its new location is to the right of the first nonzero digit. Count the number of places that the decimal point is moved. Then

1. If the original number is greater than or equal to 10:

Section 10.6 Scientific Notation

2. Converting to Scientific Notation

(continued)

Slide 103Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The exponent for the power of 10 is positive and is equal to the number of places that the decimal point was moved.

Section 10.6 Scientific Notation

2. Converting to Scientific Notation

(continued)

Slide 104Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2. If the original number is between 0 and 1:The exponent for the power of 10 is negative. Its absolute value is equal to the number of places the decimal point was moved.

Section 10.6 Scientific Notation

2. Converting to Scientific Notation

Slide 105Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

For this case, scientific notation is not needed.

3. If the original number is between 1 and 10:The exponent on 10 is 0.

Example 1 Writing Numbers in Scientific Notation

Slide 106Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Write the number in scientific notation.a. 93,000,000 mi (the distance between Earth and the Sun)b. 0.000 000 000 753 kg (the mass of a dust particle)c. 300,000,000 m/sec (the speed of light)d. 0.00017 m (length of the smallest

insect in the world)

ExampleSolution:

1 Writing Numbers in Scientific Notation

(continued)

Slide 107Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The number is greater than 10. Move the decimal point left 7 places. For a number greater than 10, the exponent is positive.

The number is between 0 and 1. Move the decimal point to the right 10 places. For a number between 0 and 1, the exponent is negative.

ExampleSolution:

1 Writing Numbers in Scientific Notation

Slide 108Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 10.6 Scientific Notation

3. Converting Scientific Notation to Standard Form

Slide 109Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

To convert from scientific notation to standard form, follow these guidelines.

1. If the exponent on 10 is positive, move the decimal point to the right the same number of places as the exponent. Add zeros as necessary.2. If the exponent on 10 is negative, move the decimal point to the left the same number of places as the exponent. Add zeros as necessary.

Example 2 Converting Scientific Notation to Standard Form

Slide 110Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Convert to decimal notation.

ExampleSolution:

2 Converting Scientific Notation to Standard Form

Slide 111Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.