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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polynomials CHAPTER 5.1Exponents and Scientific Notation 5.2Introduction to Polynomials 5.3Adding and Subtracting Polynomials 5.4Exponent Rules and Multiplying Monomials 5.5Multiplying Polynomials; Special Products 5.6Exponent Rules and Dividing Polynomials 5

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exponents and Scientific Notation 5.1 1.Evaluate exponential forms with integer exponents. 2.Write scientific notation in standard form. 3.Write standard form numbers in scientific notation.

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Slide 5- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Evaluate exponential forms with integer exponents.

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Slide 5- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluating Exponential Forms with Negative Bases If the base of an exponential form is a negative number and the exponent is even, then the product is positive. If the base is a negative number and the exponent is odd, then the product is negative. Raising a Quotient to a Power If a and b are real numbers, where b 0 and n is a natural number, then

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Slide 5- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Evaluate each exponential form. a. b. Solution a. b.

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Slide 5- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Zero as an Exponent If a is a real number and then If a is a real number, where a 0 and n is a natural number, then

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Slide 5- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rewrite with a positive exponent then, if the expression is numeric, evaluate it. a. b. Solution a. b. Rewrite using then simplify.

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Slide 5- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley If a is a real number, where a 0 and n is a natural number, then If a and b are real numbers, where a 0 and b 0 and n is a natural number, then

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Slide 5- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rewrite with a positive exponent, then evaluate it. Solution

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Slide 5- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Write scientific notation in standard form.

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Slide 5- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Scientific notation: A number expressed in the form where a is a decimal number with and n is an integer. Scientific notation gives us a shorthand way to write very large or very small numbers. Changing Scientific Notation (Positive Exponent) to Standard Form To change from scientific notation with a positive integer exponent to standard notation, move the decimal point to the right the number of places indicated by the exponent.

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Slide 5- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write in standard form. Solution Multiplying 4.26 by 10 5 means that the decimal point will move five places to the right.

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Slide 5- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Changing Scientific Notation (Negative Exponent) to Standard Form To write a number expressed in scientific notation with a negative exponent in standard form, move the decimal point to the left the same number of places as the absolute value of the exponent.

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Slide 5- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write in standard form. Solution Multiplying 3.87 by 10 –4 is equivalent to dividing by 10 4, which causes the decimal point to move four places to the left.

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Slide 5- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Write standard form numbers in scientific notation.

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Slide 5- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Changing Standard Form to Scientific Notation To write a number greater than 1 in scientific notation: 1. Move the decimal point so that the number is greater than or equal to 1, but less than 10. (Tip: Place the decimal point to the right of the first nonzero digit.) 2. Write the decimal number multiplied by 10 n, where n is the number of places between the new decimal position and the original decimal position. 3. Delete zeros to the right of the last nonzero digit.

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Slide 5- 18 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write 875,000 in scientific notation. Solution Place the decimal to the right of the first nonzero digit, 8. Next, we count the places to the right of this decimal position, which is five places. We write the 5 as the exponent of 10. Finally, we delete all the 0s to the right of the last nonzero digit, which is 5 in this case. Move the decimal point here to express a number whose absolute value is greater than or equal to 1 but less than 10. There are five places to the right of the new decimal point.

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Slide 5- 19 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Changing Standard Form to Scientific Notation To write a positive decimal number that is less than 1 in scientific notation: 1. Move the decimal point so that the number is greater than or equal to 1, but less than 10. (Tip: Place the decimal point to the right of the first nonzero digit.) 2. Write the decimal number multiplied by 10 n, where n is a negative integer whose absolute value is the number of places between the new decimal position and the original decimal position.

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Slide 5- 20 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write 0.0000000472 in scientific notation. Solution Place the decimal point between the 4 and 7 digits, so that we have 4.72, which is a decimal number greater than 1 but less than 10. Since there are eight decimal places between the original decimal position and the new position, the exponent is –8. Move the decimal point here to express a number whose absolute value is greater than or equal to 1 but less than 10. There are eight decimal places in between the original position and the new position, so the exponent is –8.

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Slide 5- 21 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate –2 –6. a) b) c) d) 64

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Slide 5- 22 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate –2 –6. a) b) c) d) 64

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Slide 5- 23 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write 13,030,000 in scientific notation. a) b) c) d)

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Slide 5- 24 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write 13,030,000 in scientific notation. a) b) c) d)

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Slide 5- 25 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write in standard notation. a) 4,690,000 b) 4,690,000,000 c) 0.000000469 d) 0.00000469

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Slide 5- 26 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write in standard notation. a) 4,690,000 b) 4,690,000,000 c) 0.000000469 d) 0.00000469

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Introduction to Polynomials 5.2 1.Identify monomials. 2.Identify the coefficient and degree of a monomial. 3.Classify polynomials. 4.Identify the degree of a polynomial. 5.Evaluate polynomials. 6.Write polynomials in descending order of degree. 7.Combine like terms.

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Slide 5- 28 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Identify monomials.

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Slide 5- 29 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Monomial: An expression that is a constant, or a product of a constant and variables that are raised to whole-number powers.

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Slide 5- 30 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Is the given expression a monomial? Explain. a. 18b. –0.4a 2 bc. 5a 2 + 4b – 1 Answer a. 18 is a monomial because it is a constant. b. –0.4a 2 b is a monomial because it is a product of a constant, –0.4, and variables, a 2 and b, which have whole-number exponents. c. 5a 2 + 4b – 1 is not a monomial because it is not a product of a constant and variables. Instead, addition and subtraction are involved.

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Slide 5- 31 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Identify the coefficient and degree of a monomial.

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Slide 5- 32 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Coefficient: The numerical factor in a monomial. Degree: The sum of the exponents of all variables in a monomial.

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Slide 5- 33 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Identify the coefficient and degree of each monomial. a. b. 9 Answer a. We can express as. In this form, we can see that our coefficient is –2 and the exponents for the variables are 1 and 3. Since the degree is the sum of the variables’ exponents, the degree is 4. b. Since 9 = 9x 0, where x is any real number except 0, we can see that 9 is the coefficient and 0 is the degree.

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Slide 5- 34 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Classify polynomials.

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Slide 5- 35 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polynomial: A monomial or an expression that can be written as a sum of monomials. Examples: 4x, 4x + 8, 2x 2 + 5xy + 8y Polynomial in one variable: A polynomial in which every variable term has the same variable. Example: x 2 – 5x + 2 is a polynomial in one variable Binomial: A polynomial containing two terms. Trinomial: A polynomial containing three terms.

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Slide 5- 36 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Determine whether the expression is a monomial, binomial, trinomial, or none of these. a. 4ab 2 b. –9x 2 + zc. 4n 3 + 2n – 1 Answer a. 4ab 2 is a monomial because it has a single term. b. –9x 2 + z is a binomial because it contains two terms. c. 4n 3 + 2n – 1 is a trinomial because it contains three terms.

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Slide 5- 37 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Identify the degree of a polynomial.

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Slide 5- 38 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Identify the degree of the polynomial. Answer To determine the degree of the monomial –2a 5 b 2, add the exponents of its variables: 5 + 2 = 7 Add the exponents of the variables in ab 2 : 1 + 2 = 3 Compare the degrees of all the terms: 4, 7, 3, 1 7 is the greatest degree. Therefore, 7 is the degree of the polynomial.

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Slide 5- 39 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 5 Evaluate polynomials.

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Slide 5- 40 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Evaluate when c = –6. Solution Replace c with –6. Simplify.

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Slide 5- 41 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 6 Write polynomials in descending order of degree.

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Slide 5- 42 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Writing a Polynomial in Descending Order To write a polynomial in descending order, place the highest degree term first, then the next highest degree, and so on.

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Slide 5- 43 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write the polynomial in descending order. Solution Rearrange the terms so that the highest degree term is first, then the next highest degree, and so on. Degree 4Degree 3Degree 2Degree 1Degree 0 Answer

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Slide 5- 44 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 7 Combine like terms.

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Slide 5- 45 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Combine like terms and write the resulting polynomial in descending order. Solution

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Slide 5- 46 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Alternative Solution Instead of first collecting like terms, we strike through like terms in the given polynomial as they are combined.

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Slide 5- 47 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Classify the expression a) Monomial b) Binomial c) Trinomial d) None of these

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Slide 5- 48 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Classify the expression a) Monomial b) Binomial c) Trinomial d) None of these

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Slide 5- 49 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate when x = –3. a) –118 b) –10 c) 10 d) 134

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Slide 5- 50 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate when x = –3. a) –118 b) –10 c) 10 d) 134

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Slide 5- 51 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Identify the degree of the polynomial. a) 3 b) 5 c) 6 d) 7

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Slide 5- 52 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Identify the degree of the polynomial. a) 3 b) 5 c) 6 d) 7

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding and Subtracting Polynomials 5.3 1.Add polynomials. 2.Subtract polynomials.

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Slide 5- 54 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Add polynomials.

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Slide 5- 55 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley We can add and subtract polynomials in the same way that we add and subtract numbers. In fact, polynomials are like whole numbers that are in an expanded form. In our base-ten number system, each place value is a power of 10. We can think of polynomials as a variable-base number system, where x 2 is like the hundreds place (10 2 ) and x is like the tens place (10 1 ). To add whole numbers, we add the digits in like place values; in polynomials, we add like terms. Adding Polynomials To add polynomials, combine like terms.

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Slide 5- 56 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Add. Solution

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Slide 5- 57 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write an expression in simplest form for the perimeter of the rectangle shown. 5b + 2 9b – 10 Understand Perimeter means the total distance around the shape. Therefore, we need to add the lengths of all the sides. Plan The lengths of the sides are represented by polynomials. Therefore, we add the polynomials to represent the perimeter.

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Slide 5- 58 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Execute Perimeter = Length + Width + Length + Width Answer The expression for the perimeter is.

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Slide 5- 59 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Check To check: 1. Choose a value for b and evaluate the original expressions for length and width. 2. Determine the corresponding numeric perimeter. 3. Evaluate the perimeter expression using the same value for b and verify that we get the same numeric perimeter. Let’s choose b = 2. Length:Width: Perimeter 408 8 12 + + + =

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Slide 5- 60 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Now evaluate the perimeter expression where b = 2 and we should find that the result is 40. Perimeter expression: This agrees with our calculation above.

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Slide 5- 61 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Subtract polynomials.

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Slide 5- 62 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Subtracting Polynomials To subtract polynomials: 1. Write the subtraction statement as an equivalent addition statement. a. Change the operation symbol from a minus sign to a plus sign. b. Change the subtrahend (second polynomial) to its additive inverse. To get the additive inverse, we change the sign of each term in the polynomial. 2. Combine like terms.

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Slide 5- 63 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Subtract. Solution Change the minus sign to a plus sign. Change all signs in the subtrahend.

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Slide 5- 64 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Add and write the polynomial in descending order. a) b) c) d)

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Slide 5- 65 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Add and write the polynomial in descending order. a) b) c) d)

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Slide 5- 66 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Subtract and write the polynomial in descending order. a) b) c) d)

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Slide 5- 67 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Subtract and write the polynomial in descending order. a) b) c) d)

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exponent Rules and Multiplying Monomials 5.4 1.Multiply monomials. 2.Multiply numbers in scientific notation. 3.Simplify a monomial raised to a power.

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Slide 5- 69 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Multiply monomials.

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Slide 5- 70 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Consider 2 3 2 4, which is a product of exponential forms. To simplify 2 3 2 4, we could follow the order of operations and evaluate the exponential forms first, then multiply. 2 3 2 4 = 8 16 = 128 However, there is an alternative. We can write the result in exponential form by first writing 2 3 and 2 4 in their factored forms. 2 3 means three 2s. 2 4 means four 2s. Since there are a total of seven 2s multiplied, we can express the product as 2 7. Notice that 2 7 = 128.

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Slide 5- 71 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Product Rule for Exponents If a is a real number and m and n are integers, then a m a n = a m+n. Multiplying Monomials To multiply monomials: 1. Multiply coefficients. 2. Add the exponents of the like bases. 3.Write any unlike variable bases unchanged in the product.

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Slide 5- 72 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. a. b. Solution a. b. 1 3 Multiply the coefficients and add the exponents of the like bases.

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Slide 5- 73 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write an expression in simplest form for the volume of the box shown. Understand We are given a box with side lengths that are monomial expressions. 3b3b 5b5b b Plan The volume of a box is found by multiplying the length, width, and height.

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Slide 5- 74 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Execute Answer The expression for volume is 15b 3. Check Since is an identity, substitute 2 for b and solve.

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Slide 5- 75 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Multiply numbers in scientific notation.

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Slide 5- 76 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley We multiply numbers in scientific notation using the same procedure we used to multiply monomials. Monomials:Scientific notation:

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Slide 5- 77 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. Write the answer in scientific notation. Solution Multiply 4.5 and 5.7, then add the exponents for base 10s. Note: The product is not in scientific notation. We must move the decimal point one place to the left and account for this by increasing the exponent by one.

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Slide 5- 78 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Simplify a monomial raised to a power.

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Slide 5- 79 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A Power Raised to a Power If a is a real number and m and n are integers, then (a m ) n = a mn. Raising a Product to a Power If a and b are real numbers and n is an integer, then (ab) n = a n b n. Simplifying a Monomial Raised to a Power To simplify a monomial raised to a power: 1. Evaluate the coefficient raised to that power. 2. Multiply each variable’s exponent by the power.

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Slide 5- 80 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. Solution Write the coefficient, 7, raised to the 4 th power, and multiply the variables’ exponent by 4. Simplify.

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Slide 5- 81 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. Solution Since the order of operations is to simplify exponents before multiplying, we will simplify (8y 7 ) 3 first, then multiply the result by 6y 2. Multiply coefficients and add exponents of like variables. Simplify.

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Slide 5- 82 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 5- 83 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 5- 84 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 5- 85 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying Polynomials; Special Products 5.5 1.Multiply a polynomial by a monomial. 2.Multiply binomials. 3. Multiply polynomials. 4.Determine the product when given special polynomial factors.

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Slide 5- 87 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Multiply a polynomial by a monomial.

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Slide 5- 88 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying a Polynomial by a Monomial To multiply a polynomial by a monomial, use the distributive property to multiply each term in the polynomial by the monomial.

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Slide 5- 89 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. a. Solution a. 2p 6p 2 2p 2p –1

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Slide 5- 90 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued b. Solution b. Note: When multiplying multivariable terms, it is helpful to multiply the coefficients first, then the variables in alphabetical order. –2a 2 b 3a 3 b –2a 2 b ab 3 –2a 2 b – 5a 4 –2a 2 b 4bc

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Slide 5- 91 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Multiply binomials.

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Slide 5- 92 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying Polynomials To multiply two polynomials: 1. Multiply every term in the second polynomial by every term in the first polynomial. 2. Combine like terms.

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Slide 5- 93 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. Solution Multiply each term in x – 5 by each term in 2x + 1 (think FOIL). FirstOuterInner Last 1 x Inner 1 (–5) Last 2x (–5) Outer 2x x First

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Slide 5- 94 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued

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Slide 5- 95 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. Solution Multiply each term in 2x + 7 by each term in 3x – 5 (think FOIL). 3x 2x 3x 7 (–5) 2x (–5) 7 FirstOuterInnerLast

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Slide 5- 96 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued

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Slide 5- 97 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The product of two binomials can be shown in terms of geometry. 35 5x5x 7x7xx2x2 Length width = Sum of the areas of the four internal rectangles Combine like terms.

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Slide 5- 98 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Multiply polynomials.

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Slide 5- 99 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. Solution Multiply each term in 2x 2 + 3x +3 by each term in x – 3. (–3) 2x 2 (–3) 3x (–3) 3 x 2x 2 x 3x x 3

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Slide 5- 100 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Determine the product when given special polynomial factors.

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Slide 5- 101 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying Conjugates If a and b are real numbers, variables, or expressions, then (a + b)(a – b) = a 2 – b 2. Conjugates: Binomials that differ only in the sign separating the terms. x + 9 and x – 9 2x + 3 and 2x – 3 –6x + 5 and –6x – 5

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Slide 5- 102 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. a. (x + 5) (x – 5) b. Solution Use (a + b)(a – b) = a 2 – b 2. Simplify. Solution Use (a + b)(a – b) = a 2 – b 2. Simplify.

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Slide 5- 103 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Squaring a Binomial (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2

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Slide 5- 104 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. Solution Use (a – b) 2 = a 2 – 2ab + b 2. Simplify.

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Slide 5- 105 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply. a) b) c) d)

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Slide 5- 106 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply. a) b) c) d)

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Slide 5- 107 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply. a) b) c) d)

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Slide 5- 108 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply. a) b) c) d)

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exponent Rules and Dividing Polynomials 5.6 1.Divide exponential forms with the same base. 2.Divide numbers in scientific notation. 3. Divide monomials. 4.Divide a polynomial by a monomial. 5.Use long division to divide polynomials. 6.Simplify expressions using rules of exponents.

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Slide 5- 110 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Divide exponential forms with the same base.

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Slide 5- 111 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quotient Rule for Exponents If m and n are integers and a is a real number, where a 0, then

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Slide 5- 112 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide. Solution Because the exponential forms have the same base, we can subtract the exponents and keep the same base.

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Slide 5- 113 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide and write the result with a positive exponent. Solution Subtract the exponents and keep the same base. Rewrite the subtraction as addition. Simplify. Write with a positive exponent.

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Slide 5- 114 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Divide numbers in scientific notation.

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Slide 5- 115 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide and write the result in scientific notation. Solution The decimal factors and powers of 10 can be separated into a product of two fractions, allowing separate division. Note: This is not in scientific notation. We must move the decimal point one place to the right and account for this by decreasing the exponent by one.

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Slide 5- 116 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Divide monomials.

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Slide 5- 117 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing Monomials To divide monomials: 1. Divide the coefficients. 2. Use the quotient rule for the exponents with like bases. 3.Unlike bases are written unchanged in the quotient. 4.Write the final expression so that all exponents are positive.

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Slide 5- 118 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide. Solution

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Slide 5- 119 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Divide a polynomial by a monomial.

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Slide 5- 120 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley If a, b, and c are real numbers, variables, or expressions with c 0, then Dividing a Polynomial by a Monomial To divide a polynomial by a monomial, divide each term in the polynomial by the monomial.

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Slide 5- 121 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide. Solution Divide each term in the polynomial by the monomial.

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Slide 5- 122 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 5 Use long division to divide polynomials.

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Slide 5- 123 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To divide a polynomial by a polynomial, we can use long division. Divide: Divisor Quotient Remainder Quotient 13 Divisor 12 Remainder 1 = Dividend =157 + +

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Slide 5- 124 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing a Polynomial by a Polynomial To divide a polynomial by a polynomial, use long division. If there is a remainder, write the result in the following form:

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Slide 5- 125 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide. Solution Begin by dividing the first term in the dividend by the first term in the divisor: 6b 2 + 5b – 28. Change signs.

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Slide 5- 126 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Determine the next part of the quotient by dividing 8b by 2b, which is 4, and repeat the multiplication and subtraction steps. Change signs. Answer

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Slide 5- 127 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 6 Simplify expressions using rules of exponents.

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Slide 5- 128 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exponents Summary Assume that no denominators are 0, that a and b are real numbers, and that m and n are integers. Zero as an exponent:a 0 = 1, where a 0. 0 0 is indeterminate. Negative exponents: Product rule for exponents: Quotient rule for exponents: Raising a power to a power: Raising a product to a power: Raising a quotient to a power:

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Slide 5- 129 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. Write all answers with positive exponents. a. Solution a.

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Slide 5- 130 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued b. Solution Use the quotient rule for exponents. Write with a positive exponent.

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Slide 5- 131 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 5- 132 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 5- 133 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Divide. a) b) c) d)

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Slide 5- 134 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Divide. a) b) c) d)

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Polynomials and Polynomial Functions

Polynomials and Polynomial Functions

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