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Transcript of CHAPTER OUTLINE 10 Exponents and Polynomials Slide 2 Copyright (c) The McGraw-Hill Companies, Inc....
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
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1. Key Definitions2. Addition of Polynomials3. Subtraction of Polynomials4. Evaluating Polynomials and Applications
Section 10.1 Addition and Subtraction of Polynomials
1. Key Definitions
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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
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• 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
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This is written in descending order.
The degree is 6.
Example 1 Identifying the Characteristics of a Polynomial
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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
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Section 10.1 Addition and Subtraction of Polynomials
1. Key Definitions
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The coefficient of a term is the numerical factorof the term.
Section 10.1 Addition and Subtraction of Polynomials
2. Addition of Polynomials
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Like terms have the same variables, raised to the same powers.
Example 2 Combining Like Terms
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ExampleSolution:
2 Combining Like Terms
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Section 10.1 Addition and Subtraction of Polynomials
2. Addition of Polynomials
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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
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ExampleSolution:
3 Adding Polynomials
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TIP:
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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
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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
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Write the opposite of the polynomial.
ExampleSolution:
5 Finding the Opposite of a Polynomial
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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
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ExampleSolution:
6 Subtracting Polynomials
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Write the opposite of the second polynomial by applying the distributive property.
Regroup and collect like terms.
Combine like terms.
TIP:
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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
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ExampleSolution:
7 Subtracting Polynomials
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Section 10.1 Addition and Subtraction of Polynomials
4. Evaluating Polynomials and Applications
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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
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Evaluate the polynomial for the given value of the variable:
ExampleSolution:
8 Evaluating a Polynomial
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Example 9 Evaluating a Polynomial in an Application
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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
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Section
Objectives
10.2 Multiplication Properties of Exponents
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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
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Example 1 Multiplying Factors with the Same Base
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ExampleSolution:
1 Multiplying Factors with the Same Base
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The base remains unchanged. Add the exponents.
Section 10.2 Multiplication Properties of Exponents
2. Multiplying Monomials
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We use the commutative and associative properties of multiplication to regroup factors and multiply like bases.
Example 2 Multiplying Monomials
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ExampleSolution:
2 Multiplying Monomials
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Example 3 Multiplying Monomials
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ExampleSolution:
3 Multiplying Monomials
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PROPERTY Power Rule for Exponents
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Example 4 Applying the Power Rule for Exponents
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Simplify the expressions.
ExampleSolution:
4 Applying the Power Rule for Exponents
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Avoiding Mistakes
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PROPERTY Power of a Product and the Power of a Quotient
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Example 5 Simplifying a Power of a Product or Quotient
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Simplify.
ExampleSolution:
5 Simplifying a Power of a Product or Quotient
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Example 6 Simplifying Expressions Involving Exponents
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ExampleSolution:
6 Simplifying Expressions Involving Exponents
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Apply the power rule to each factor in parentheses. Operations with exponents are performed before multiplication.
Section
Objectives
10.3 Multiplication of Polynomials
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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
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To multiply a monomial by a polynomial with more than one term, use the distributive property.
Example 1 Multiplying a Monomial by a Polynomial
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ExampleSolution:
1 Multiplying a Monomial by a Polynomial
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Example 2 Multiplying a Monomial by a Polynomial
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ExampleSolution:
2 Multiplying a Monomial by a Polynomial
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PROCEDURE Multiplying Polynomials
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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
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ExampleSolution:
4 Multiplying a Binomial by a Binomial
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Multiply each term in the first binomial by each term in the second binomial.
TIP:
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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
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ExampleSolution:
5 Multiplying a Binomial by a Binomial
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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
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ExampleSolution:
7 Squaring a Binomial
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To square a quantity, multiplythe quantity times itself.
Example 8 Multiplying a Polynomial by a Polynomial
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ExampleSolution:
8 Multiplying a Polynomial by a Polynomial
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Apply the distributive property. Multiply each term in the first polynomial by each term in the second.
Section
Objectives
10.4 Introduction to Factoring
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1. Greatest Common Factor2. Factoring Out the Greatest Common Factor
Section 10.4 Introduction to Factoring
1. Greatest Common Factor
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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
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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
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Determine the GCF.
ExampleSolution:
1 Determining the Greatest Common Factor
(continued)
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ExampleSolution:
1 Determining the Greatest Common Factor
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Example 2 Determining the Greatest Common Factor
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Determine the GCF.
ExampleSolution:
2 Determining the Greatest Common Factor
(continued)
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ExampleSolution:
2 Determining the Greatest Common Factor
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Example 3 Factoring Out the GCF
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ExampleSolution:
3 Factoring Out the GCF
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Example 5 Factoring Out the GCF
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ExampleSolution:
5 Factoring Out the GCF
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Section
Objectives
10.5 Negative Exponents and the Quotient Rule for Exponents
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1. Division of Like Bases2. Definition of b0
3. Definition of b–n
4. Properties of Exponents: A Summary
PROPERTY Division of Like Bases
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Example 1 Dividing Like Bases
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ExampleSolution:
1 Dividing Like Bases
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The base is unchanged. Subtract the exponents.
DEFINITION b0
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Let b be a nonzero number. Then b0 = 1.
Example 2 Simplifying Expressions with a Zero Exponent
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ExampleSolution:
2 Simplifying Expressions with a Zero Exponent
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DEFINITION b –n
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Section 10.5 Negative Exponents and the Quotient Rule for Exponents
3. Definition of b–n
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To evaluate b–n, take the reciprocal of the base and change the sign of the exponent.
Example 3 Simplifying Expressions Containing Negative Exponents
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ExampleSolution:
3 Simplifying Expressions Containing Negative Exponents
(continued)
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Take the reciprocal of the base. Change the sign of the exponent.
ExampleSolution:
3 Simplifying Expressions Containing Negative Exponents
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Example 4 Simplifying Expressions Containing Negative Exponents
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ExampleSolution:
4 Simplifying Expressions Containing Negative Exponents
(continued)
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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
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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
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ExampleSolution:
5 Simplifying Expressions ContainingNegative Exponents
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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
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Example 6 Simplifying Expressions Containing Exponents
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ExampleSolution:
6 Simplifying Expressions Containing Exponents
(continued)
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The base is unchanged. Add the exponents (property 1).
The base is unchanged. Subtract the exponents (property 2).
ExampleSolution:
6 Simplifying Expressions Containing Exponents
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Section
Objectives
10.6 Scientific Notation
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1. Scientific Notation2. Converting to Scientific Notation3. Converting Scientific Notation to Standard Form
Section 10.6 Scientific Notation
1. Scientific Notation
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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
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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
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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)
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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)
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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)
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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
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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
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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)
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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
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Section 10.6 Scientific Notation
3. Converting Scientific Notation to Standard Form
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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
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Convert to decimal notation.