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Chapter Nine Section Three Multiplying a Polynomial by a Monomial

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What You’ll Learn You’ll learn to multiply a polynomial by a monomial.

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Why it’s Important Recreation You can use polynomials to solve problems involving recreation.

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Suppose you have a square whose length and width are x feet. If you increase the length by 3 feet, what is the area of the new figure? You can model this problem by using algebra tiles. The figures on the next slide show how to make a rectangle whose length is x + 3 feet and whose width is x feet.

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The area of any rectangle is the product of its length and its width. The area can be found by adding the areas of the tiles. x2x2 xxxx2x2 xxx x x + 3 ft. x x ft. 111

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Formula A = lw A = (x + 3)x or x(x + 3) A = x 2 + 3x Since the areas are equal, x(x + 3) = x 2 + 3x square feet. The example above shows how the Distributive Property can be used to multiply a polynomial by a monomial. x2x2 xxxx2x2 xxx x x + 3 ft. x x ft. 111

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Multiplying a Polynomial by a Monomial Words: To multiplying a polynomial by a monomial, use the Distributive Property. Symbols: a (b + c) = ab + ac Model: b c a abac

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Example One Find each product. y(y + 5) y(y + 5) = y(y) + y(5) = y 2 + 5y 5yy2y2 y y5

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Example Two Find each product. b(2b 2 + 3) b(2b 2 + 3)= b(2b 2 ) + b(3) = 2b 3 + 3b 3b2b 3 b 2b 2 +3

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Example Three Find each product. -2n(7 – 5n 2 ) -2n(7 – 5n 2 ) = -2n(7) + -2n(5n 2 ) = -14n + 10n 3 10n 3 -14n -2n 7-5n 2

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Example Four Find each product. 3x 3 (2x 2 – 5x + 8) 3x 3 (2x 2 – 5x + 8) = 3x 3 (2x 2 ) + 3x 3 (-5x) + 3x 3 (8) = 6x 5 – 15x 4 + 24x 3

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Your Turn Find each product. 7(2x + 5) 14x + 35

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Your Turn Find each product. 4x(3x 2 - 7) 12x 3 - 28x

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Your Turn Find each product. -5a(6 – 3a 2 ) -30a + 15a 3

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Your Turn Find each product. 2m 2 (5m 2 – 7m + 8) 10m 4 – 14m 3 + 16m 2

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Example Five Many equations contain polynomials that must be multiplied. Solve each equation. 11(y -3) + 5 = 2(y +22) 11y – 33 + 5 = 2y + 44 Distributive Property 11y – 28 = 2y + 44 Combine Like Terms -2y -2y Subtract 2y from each side 9y – 28 = 44 + 28 +28 Add 28 to each side. 9y = 72 Divide each side by 9 9 Y = 8

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Example Six Solve each equation. w(w + 12) = w(w + 14) + 12 w(w + 12) = w(w + 14) + 12 Distributive Property w 2 + 12w = w 2 + 14w + 12 -w 2 -w 2 Subtract w 2 from each side 12w = 14w + 12 -14w -14w Subtract 14w to each side. -2w = 12 Divide each side by -2 -2 w = -6

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Your Turn Solve each equation. 2(5x - 12) = 6(-2x + 3) + 2 2

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Your Turn Solve each equation. a(a + 2) + 3a = a(a - 3) + 8 1

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