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Copyright © 2010 Pearson Education, Inc
Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 1
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4 Polynomials 4.1 Exponents and Their Properties
4.2 Negative Exponents and Scientific Notation 4.3 Polynomials 4.4 Addition and Subtraction of Polynomials 4.5 Multiplication of Polynomials 4.6 Special Products 4.7 Polynomials in Several Variables 4.8 Division of Polynomials Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Exponents and Their Products
4.1 Exponents and Their Products Multiplying Powers with Like Bases Dividing Powers with Like Bases Zero as an Exponent Raising a Power to a Power Raising a Product or a Quotient to a Power Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The Product Rule For any number a and any positive integers m and n, (To multiply powers with the same base, keep the base and add the exponents.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a) x3 x5 b) 62 67 63 c) (x + y)6(x + y)9 d) (w3z4)(w3z7) Solution a) x3 x5 = x Adding exponents = x8 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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continued Example b) 62 67 63 c) (x + y)6(x + y)9 d) (w3z4)(w3z7)
Solution b) 62 67 63 = = 612 c) (x + y)6(x + y)9 = (x + y)6+9 = (x + y)15 d) (w3z4)(w3z7) = w3z4w3z7 = w3w3z4z7 = w6z11 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The Quotient Rule For any nonzero number a and any positive integers m and n for which m > n, (To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Divide and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a) b) c) d) Solution a) b) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example continued c) d) Solution c) d)
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The Exponent Zero (Any nonzero number raised to the 0 power is 1.)
For any real number a, with a ≠ 0, (Any nonzero number raised to the 0 power is 1.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Simplify: a) 12450 b) (3)0 c) (4w)0 d) (1)80 e) 80.
Solution a) = 1 b) (3)0 = 1 c) (4w)0 = 1, for any w 0. d) (1)80 = (1)1 = 1 e) 80 is read “the opposite of 90” and is equivalent to (1)90: 90 = (1)90 = (1)1 = 1 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The Power Rule (am)n = amn.
For any number a and any whole numbers m and n, (am)n = amn. (To raise a power to a power, multiply the exponents and leave the base unchanged.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Simplify: a) (x3)4 b) (42)6 Solution a) (x3)4 = x34 = x12
= 416 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Raising a Product to a Power
For any numbers a and b and any whole number n, (ab)n = anbn. (To raise a product to a power, raise each factor to that power.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Simplify: a) (3x)4 b) (2x3)2 c) (a2b3)7(a4b5) Solution
a) (3x)4 = 34x4 = 81x4 b) (2x3)2 = (2)2(x3)2 = 4x6 c) (a2b3)7(a4b5) = (a2)7(b3)7a4b5 = a14b21a4b Multiplying exponents = a18b Adding exponents Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Raising a Quotient to a Power
For any real numbers a and b, b ≠ 0, and any whole number n, (To raise a quotient to a power, raise the numerator to the power and divide by the denominator to the power.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Simplify: a) b) c) Solution a) b)
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continued Example Simplify: c) Solution c)
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Definitions and Properties of Exponents For any whole numbers m and n,
1 as an exponent: a1 = a 0 as an exponent: a0 = 1 The Product Rule: The Quotient Rule: The Power Rule: (am)n = amn Raising a product to a power: (ab)n = anbn Raising a quotient to a power: Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Negative Exponents and Scientific Notation
4.2 Negative Exponents and Scientific Notation Negative Integers as Exponents Scientific Notation Multiplying and Dividing Using Scientific Notation Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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For any real number a that is nonzero and any integer n,
Negative Exponents For any real number a that is nonzero and any integer n, (The numbers a-n and an are reciprocals of each other.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Express using positive exponents, and, if possible, simplify.
a) m5 b) 52 c) (4)2 d) xy1 Solution a) m5 = b) 52 = Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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continued Example c) (4)2 = d) xy1 =
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Example Simplify. Do not use negative exponents in the answer.
a) b) (x4)3 c) (3a2b4)3 d) e) f) Solution a) b) (x4)3 = x(4)(3) = x12 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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continued Example c) (3a2b4)3 = 33(a2)3(b4) = 27 a6b12 = d) e) f)
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Factors and Negative Exponents
For any nonzero real numbers a and b and any integers m and n, (A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Simplify. Solution
We can move the negative factors to the other side of the fraction bar if we change the sign of each exponent. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Reciprocals and Negative Exponents
For any nonzero real numbers a and b and any integer n, (Any base to a power is equal to the reciprocal of the base raised to the opposite power.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Simplify: Solution
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Scientific Notation N × 10m,
Scientific notation for a number is an expression of the type N × 10m, where N is at least 1 but less than 10 (that is, 1 ≤ N < 10), N is expressed in decimal notation, and m is an integer. Note that when m is positive the decimal point moves right m places in decimal notation. When m is negative, the decimal point moves left |m| places. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Convert to decimal notation: a) 3.842 106 b) 5.3 107
Solution a) Since the exponent is positive, the decimal point moves right 6 places. 106 = 3,842,000 b) Since the exponent is negative, the decimal point moves left 7 places. 107 = Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Write in scientific notation: a) 94,000 b) 0.0423 Solution
a) We need to find m such that 94, = 9.4 10m. This requires moving the decimal point 4 places to the right. 94,000 = 9.4 104 b) To change 4.23 to we move the decimal point 2 places to the left. = 4.23 102 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Multiplying and Dividing Using Scientific Notation
Products and quotients of numbers written in scientific notation are found using the rules for exponents. Simplify: (1.7 108)(2.2 105) Solution (1.7 108)(2.2 105) = 1.7 2.2 108 105 = 3.74 108 +(5) = 3.74 103 Example Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Simplify. (6.2 109) (8.0 108) Solution
(6.2 109) (8.0 108) = Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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4.3 Polynomials Terms Types of Polynomials Degree and Coefficients
Combining Like Terms Evaluating Polynomials and Applications Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Terms A term can be a number, a variable, a product of numbers and/or variables, or a quotient of numbers and/or variables. A term that is a product of constants and/or variables is called a monomial. Examples of monomials: 8, w, x3y A polynomial is a monomial or a sum of monomials. Examples of polynomials: 5w + 8, 3x2 + x + 4, x, 0, y6 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Identify the terms of the polynomial 7p5 3p3 + 3. Solution
The terms are 7p5, 3p3, and 3. We can see this by rewriting all subtractions as additions of opposites: 7p5 3p3 + 3 = 7p5 + (3p3) + 3 These are the terms of the polynomial. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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A polynomial that is composed of two terms is called a binomial, whereas those composed of three terms are called trinomials. Polynomials with four or more terms have no special name. Monomials Binomials Trinomials No Special Name 5x2 3x + 4 3x2 + 5x + 9 5x3 6x2 + 2xy 9 8 4a5 + 7bc 7x7 9z3 + 5 a4 + 2a3 a2 + 7a 2 8a23b3 10x3 7 6x2 4x ½ 6x6 4x5 + 2x4 x3 + 3x 2 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The degree of a term of a polynomial is the number of variable factors in that term.
Determine the degree of each term: a) 9x5 b) 6y c) 9 Solution a) The degree of 9x5 is 5. b) The degree of 6y is 1. c) The degree of 9 is 0. Example Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The part of a term that is a constant factor is the coefficient of that term. The coefficient of 4y is 4. Identify the coefficient of each term in the polynomial. 5x4 8x2y + y 9 Solution The coefficient of 5x4 is 5. The coefficient of 8x2y is 8. The coefficient of y is 1, since y = 1y. The coefficient of 9 is simply 9. Example Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The terms are 4x2, 9x3, 6x4, 8x, and 7.
The leading term of a polynomial is the term of highest degree. Its coefficient is called the leading coefficient and its degree is referred to as the degree of the polynomial. Consider this polynomial 4x2 9x3 + 6x4 + 8x 7. The terms are x2, 9x3, 6x4, 8x, and 7. The coefficients are , 9, 6, and 7. The degree of each term is 2, 3, , 1, and 0. The leading term is 6x4 and the leading coefficient is 6. The degree of the polynomial is 4. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Combine like terms. a) 4y4 9y4
b) 7x x2 + 6x2 13 6x5 c) 9w5 7w3 + 11w5 + 2w3 Solution a) 4y4 9y4 = (4 9)y4 = 5y4 b) 7x x2 + 6x2 13 6x5 = 7x5 6x5 + 3x2 + 6x2 + 9 13 = x5 + 9x2 4 c) 9w5 7w3 + 11w5 + 2w3 = 9w5 + 11w5 7w3 + 2w3 = 20w5 5w3 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Evaluate x3 + 4x + 7 for x = 3. Solution For x = 3, we have
= (27) + 4(3) + 7 = 27 + (12) + 7 = 22 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example In a sports league of n teams in which each team plays every other team twice, the total number of games to be played is given by the polynomial n2 n. A boys’ soccer league has 12 teams. How many games are played if each team plays every other team twice? Solution We evaluate the polynomial for n = 12: n2 n = 122 12 = 144 12 = 132. The league plays 132 games. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Solution 0.4r2 40r + 1039 = 0.4(25)2 40(25) + 1039
The average number of accidents per day involving drivers of age r can be approximated by the polynomial 0.4r2 40r Find the average number of accidents per day involving 25-year-old drivers. Solution 0.4r2 40r = 0.4(25)2 40(25) = 0.4(625) = 250 = 289 There are, on average, approximately 289 accidents each day involving 25-year-old drivers. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Addition and Subtraction of Polynomials
4.4 Addition and Subtraction of Polynomials Addition of Polynomials Opposites of Polynomials Subtraction of Polynomials Problem Solving Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Add. (6x3 + 7x 2) + (5x3 + 4x2 + 3) Solution
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Example Add: (3 4x + 2x2) + (6 + 8x 4x2 + 2x3) Solution
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Example Add: 10x5 3x3 + 7x2 + 4 and 6x4 8x2 + 7 and 4x6 6x5 + 2x2 + 6 Solution 10x 3x3 + 7x2 + 4 6x 8x2 + 7 4x6 6x x2 + 6 4x6 + 4x5 + 6x4 3x3 + x2 + 17 The answer is 4x6 + 4x5 + 6x4 3x3 + x Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The Opposite of a Polynomial
To find an equivalent polynomial for the opposite, or additive inverse, of a polynomial, change the sign of every term. This is the same as multiplying the polynomial by –1. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Simplify: (8x4 x3 + 9x2 2x + 72) Solution
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Subtraction of Polynomials
We can now subtract one polynomial from another by adding the opposite of the polynomial being subtracted. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Subtract: (10x5 + 2x3 3x2 + 5) (3x5 + 2x4 5x3 4x2)
Solution = 10x5 + 2x3 3x x5 2x4 + 5x3 + 4x2 = 13x5 2x4 + 7x3 + x2 + 5 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Solution (8x5 + 2x3 10x) (4x5 5x3 + 6)
Subtract: (8x5 + 2x3 10x) (4x5 5x3 + 6) Solution (8x5 + 2x3 10x) (4x5 5x3 + 6) = 8x5 + 2x3 10x + (4x5) + 5x3 6 = 4x5 + 7x3 10x 6 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Write in columns and subtract:
(6x2 4x + 7) (10x2 6x 4) Solution 6x2 4x + 7 (10x2 6x 4) 4x2 + 2x + 11 Remember to change the signs Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example A 6-ft by 5-ft hot tub is installed on an outdoor deck measuring w ft by w ft. Find a polynomial for the remaining area of the deck. Solution 1. Familiarize. We make a drawing of the situation as follows. w ft 7 ft 5 ft Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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continued Example 2. Translate. Rewording: Area of Area of Area
deck tub = left over Translating: w ft w ft 5 ft 7 ft = Area left over 3. Carry out. w ft2 35 ft2 = Area left over. 4. Check. As a partial check, note that the units in the answer are square feet, a measure of area, as expected. 5. State. The remaining area in the yard is (x2 35)ft2. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Multiplication of Polynomials
4.5 Multiplication of Polynomials Multiplying Monomials Multiplying a Monomial and a Polynomial Multiplying Any Two Polynomials Checking by Evaluating Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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To Multiply Monomials To find an equivalent expression for the product of two monomials, multiply the coefficients and then multiply the variables using the product rule for exponents. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply: a) (6x)(7x) b) (5a)(a) c) (8x6)(3x4) Solution
a) (6x)(7x) = (6 7) (x x) = 42x2 b) (5a)(a) = (5a)(1a) = (5)(1)(a a) = 5a2 c) (8x6)(3x4) = (8 3) (x6 x4) = 24x6 + 4 = 24x10 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Solution Multiply: a) x and x + 7 b) 6x(x2 4x + 5)
a) x(x + 7) = x2 + 7x b) 6x(x2 4x + 5) = 6x3 24x2 + 30x = x x + x 7 = (6x)(x2) (6x)(4x) + (6x)(5) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The Product of a Monomial and a Polynomial
To multiply a monomial and a polynomial, multiply each term of the polynomial by the monomial. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Solution Multiply: 5x2(x3 4x2 + 3x 5)
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Example a) x + 3 and x + 5 b) 3x 2 and x 1 Solution
Multiply each of the following. a) x + 3 and x + 5 b) 3x 2 and x 1 Solution a) (x + 3)(x + 5) = (x + 3)x + (x + 5)3 = x(x + 3) + 5(x + 5) = x x + x x + 5 3 = x2 + 3x + 5x + 15 = x2 + 8x + 15 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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continued Solution b) (3x 2)(x 1) = (3x 2)x (3x 2)1
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The Product of Two Polynomials
To multiply two polynomials P and Q, select one of the polynomials, say P. Then multiply each term of P by every term of Q and combine like terms. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply: (5x3 + x2 + 4x)(x2 + 3x) Solution 5x3 + x2 + 4x
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Example Multiply: (3x2 4)(2x2 3x + 1) Solution 2x2 3x + 1
6x4 + 9x3 11x2 + 12x 4 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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4.6 Special Products Products of Two Binomials
Multiplying Sums and Differences of Two Terms Squaring Binomials Multiplications of Various Types Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The FOIL Method To multiply two binomials, A + B and C + D, multiply the First terms AC, the Outer terms AD, the Inner terms BC, and then the Last terms BD. Then combine like terms, if possible. (A + B)(C + D) = AC + AD + BC + BD Multiply First terms: AC. Multiply Outer terms: AD. Multiply Inner terms: BC Multiply Last terms: BD ↓ FOIL (A + B)(C + D) O I F L Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply: (x + 4)(x2 + 3) Solution F O I L
(x + 4)(x2 + 3) = x3 + 3x + 4x2 + 12 = x3 + 4x2 + 3x + 12 O I F L The terms are rearranged in descending order for the final answer. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply. a) (x + 8)(x + 5) b) (y + 4) (y 3)
c) (5t3 + 4t)(2t2 1) d) (4 3x)(8 5x3) Solution a) (x + 8)(x + 5) = x2 + 5x + 8x + 40 = x2 + 13x + 40 b) (y + 4) (y 3) = y2 3y + 4y 12 = y2 + y 12 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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continued Example Solution
c) (5t3 + 4t)(2t2 1) = 10t5 5t3 + 8t3 4t = 10t5 + 3t3 4t d) (4 3x)(8 5x3) = 32 20x3 24x + 15x4 = 32 24x 20x3 + 15x4 In general, if the original binomials are written in ascending order, the answer is also written that way. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The Product of a Sum and Difference
The product of the sum and difference of the same two terms is the square of the first term minus the square of the second term. (A + B)(A – B) = A2 – B2. This is called a difference of squares. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply. a) (x + 8)(x 8) b) (6 + 5w) (6 5w)
c) (4t3 3)(4t3 + 3) Solution (A + B)(A B) = A2 B2 a) (x + 8)(x 8) = x2 82 = x2 64 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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continued Example Solution b) (6 + 5w) (6 5w) = 62 (5w)2
c) (4t3 3)(4t3 + 3) = (4t3)2 32 = 16t6 9 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The Square of a Binomial
The square of a binomial is the square of the first term, plus twice the product of the two terms, plus the square of the last term. (A + B)2 = A2 + 2AB + B2; (A – B)2 = A2 – 2AB + B2; These are called perfect-square trinomials.* *In some books, these are called trinomial squares. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply. a) (x + 8)2 b) (y 7)2 c) (4x 3x5)2 Solution
(A + B)2 = A2+2AB + B2 a) (x + 8)2 = x2 + 2x8 + 82 = x2 + 16x + 64 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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continued (A + B)2 = A2 2AB + B2 Example
Solution b) (y 7)2 = y2 2 y = y2 14y + 49 c) (4x 3x5)2 = (4x)2 2 4x 3x5 + (3x5)2 = 16x2 24x6 + 9x10 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Multiplying Two Polynomials
1. Is the multiplication the product of a monomial and a polynomial? If so, multiply each term of the polynomial by the monomial. 2. Is the multiplication the product of two binomials? If so: a) Is the product of the sum and difference of the same two terms? If so, use the pattern (A + B)(A B) = (A B)2. b) Is the product the square of a binomial? If so, us the pattern (A + B)2 = A2 + 2AB + B2, or (A – B)2 = A2 – 2AB + B2. c) If neither (a) nor (b) applies, use FOIL. 3. Is the multiplication the product of two polynomials other than those above? If so, multiply each term of one by every term of the other. Use columns if you wish. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply. a) (x + 5)(x 5) b) (w 7)(w + 4)
c) (x + 9)(x + 9) d) 3x2(4x2 + x 2) e) (p + 2)(p2 + 3p 2) f) (2x + 1)2 Solution a) (x + 5)(x 5) = x2 25 b) (w 7)(w + 4) = w2 + 4w 7w 28 = w2 3w 28 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply continued c) (x + 9)(x + 9) = x2 + 18x + 81
d) 3x2(4x2 + x 2) = 12x4 + 3x3 6x2 e) p2 + 3p 2 p + 2 2p2 + 6p 4 p3 + 3p2 2p p3 + 5p2 + 4p 4 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Multiply continued f) (2x + 1)2 = 4x2 + 2(2x)(1) + 1
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Polynomials in Several Variables
4.7 Polynomials in Several Variables Evaluating Polynomials Like Terms and Degree Addition and Subtraction Multiplication Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Evaluate the polynomial x + xy2 + 9x3y for x = 3 and y = 4. Solution We substitute 3 for x and 4 for y: 5 + 4x + xy2 + 9x3y2 = 5 + 4(3) + (3)(42) + 9(3)3(4)2 = 5 12 48 3888 = 3943 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example The surface area of a right circular cylinder is given by the polynomial 2rh + 2r2 where h is the height and r is the radius of the base. A barn silo has a height of 50 feet and a radius of 9 feet. Approximate its surface area. Solution We evaluate the polynomial for h = 50 ft and r = 9 ft. If 3.14 is used to approximate , we have h r Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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continued h = 50 ft and r = 9 ft
2rh + 2r2 2(3.14)(9 ft)(50 ft) + 2(3.14)(9 ft)2 2(3.14)(9 ft)(50 ft) + 2(3.14)(81 ft2) 2826 ft ft2 ft2 Note that the unit in the answer (square feet) is a unit of area. The surface area is about ft2 (square feet). Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Recall that the degree of a monomial is the number of variable factors in the term.
Identify the coefficient and the degree of each term and the degree of the polynomial 10x3y2 15xy3z4 + yz + 5y + 3x2 + 9. Example Term Coefficient Degree Degree of the Polynomial 10x3y2 10 5 8 15xy3z4 15 yz 1 2 5y 3x2 3 9 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Like Terms Like, or similar terms either have exactly the same variables with exactly the same exponents or are constants. For example, 9w5y4 and 15w5y4 are like terms and 12 and 14 are like terms, but 6x2y and 9xy3 are not like terms. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Combine like terms. Example a) 10x2y + 4xy3 6x2y 2xy3
b) 8st 6st2 + 4st2 + 7s3 + 10st 12s3 + t 2 Solution = (10 6)x2y + (42)xy3 = 4x2y + 2xy3 = 5s3 2st2 + 18st + t 2 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Addition and Subtraction
Example Add: (6x3 + 4y 6y2) + (7x3 + 5x2 + 8y2) Solution (6x3 + 4y 6y2) + (7x3 + 5x2 + 8y2) = (6 + 7)x3 + 5x2 + 4y + (6 + 8)y2 = x3 + 5x2 + 4y + 2y2 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Subtract: (5x2y + 2x3y2 + 4x2y3 + 7y) (5x2y 7x3y2 + x2y2 6y) Solution = (5x2y + 2x3y2 + 4x2y3 + 7y) 5x2y + 7x3y2 x2y2 + 6y = 9x3y2 + 4x2y3 x2y2 + 13y Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Multiplication Example Multiply: (4x2y 3xy + 4y)(xy + 3y) Solution
4x3y2 3x2y2 + 4xy2 4x3y2 + 9x2y2 5xy2 + 12y2 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The special products discussed in Section 4.5 can speed up your work.
Example Multiply. a) (x + 6y)(2x 3y) b) (5x + 7y)2 c) (a4 5a2b2) d) (7a2b + 3b)(7a2b 3b) e) (3x3y2 + 7t)(3x3y2 + 7t) f) (3x + 1 4y)(3x y) Solution a) (x + 6y)(2x 3y) = 2x2 3xy + 12xy 18y2 = 2x2 + 9xy 18y FOIL Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Solution continued b) (5x + 7y)2 = (5x)2 + 2(5x)(7y) + (7y)2
c) (a4 5a2b2)2 = (a4)2 2(a4)(5a2b2) + (5a2b2)2 = a8 10a6b2 + 25a4b4 d) (7a2b + 3b)(7a2b 3b) = (7a2b)2 (3b)2 = 49a4b2 9b2 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Solution continued e) (3x3y2 + 7t)(3x3y2 + 7t)
= (7t 3x3y2)(7t + 3x3y2) = (7t)2 (3x3y2)2 = 49t2 9x6y4 f) (3x + 1 4y)(3x y) = (3x + 1)2 (4y)2 = 9x2 + 6x + 1 16y2 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Division of Polynomials
4.8 Division of Polynomials Dividing by a Monomial Dividing by a Binomial Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Dividing by a Monomial Example
To divide a polynomial by a monomial, we divide each term by the monomial. Divide. x5 + 24x4 12x3 by 6x Solution Example Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Divide. Solution
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Dividing by a Binomial For divisors with more than one term, we use long division, much as we do in arithmetic. Polynomial are written in descending order and any missing terms in the dividend are written in, using 0 for the coefficients. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Divide x2 + 7x + 12 by x + 3. Solution
Now we “bring down” the next term. Multiply x + 3 by x, using the distributive law Subtract by changing signs and adding Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Solution continued Multiply 4 by the divisor, x + 3, using the distributive law Subtract Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Divide 15x2 22x + 14 by 3x 2. Solution
The answer is 5x 4 with R6. Another way to write the answer is Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Divide x5 3x4 4x2 + 10x by x 3 . Solution The answer is
Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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