6.8 Multiplying Polynomials

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6.8 Multiplying Polynomials CORD Math Mrs. Spitz Fall 2006

Objectives: After studying this lesson, you should be able to: Use the FOIL method to multiply to binomials, and Multiply any two polynomials using the distributive property.

Assignment Worksheet 6.8 Reminder there is a test over chapter 9 after 6.9 and a review for the chapter.

Connection: You know that the area of a rectangle is the product of its length and width. You can multiply 2x + 3 and 5x + 8 to find the area of a large rectangle. 5x cm 8 cm 2x cm 3 cm

Connection: (2x + 3)(5x + 8) = 2x(5x + 8) + 3(5x + 8) But you also know that the area of the large rectangle equals the sum of the areas of the four smaller rectangles, don’t you?

Connection: = 10x2 + 16x + 15x + 24 = 10x2 + 31x + 24 (2x + 3)(5x + 8) = 2x · 5x + 2x ·8 + 3 · 5x + 3 ·8 = 10x2 + 16x + 15x + 24 = 10x2 + 31x + 24 This example illustrates a shortcut of the distributive property called the FOIL METHOD.

(2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15 F.O.I.L. There is an acronym to help us remember how to multiply two binomials without stacking them. (2x + -3)(4x + 5) F : Multiply the First term in each binomial. 2x • 4x = 8x2 O : Multiply the Outer terms in the binomials. 2x • 5 = 10x I : Multiply the Inner terms in the binomials. -3 • 4x = -12x L : Multiply the Last term in each binomial. -3 • 5 = -15 (2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15

Use the FOIL method to multiply these binomials: 1) (3a + 4)(2a + 1) 2) (x + 4)(x - 5) 3) (x + 5)(x - 5) 4) (c - 3)(2c - 5) 5) (2w + 3)(2w - 3)

Use the FOIL method to multiply these binomials: 6a2 + 3a + 8a + 4 = 6a2 + 11a + 4 x2 + -5x + 4x + -20 = x2 - x -20 x2 + -5x + 5x + -25 = x2 – 25 2c2 + -5x + - 6x + 15 = 2c2 -11x +15 4w2 + -6w + 6w + -9 = 4w2 - 9 (3a + 4)(2a + 1) = (x + 4)(x - 5) = (x + 5)(x - 5) = (c - 3)(2c - 5) = (2w + 3)(2w - 3) =

Ex. 1 Find (y + 5)(y + 7) Ex. 2 Find (3x - 5)(5x + 2) (y + 5)(y + 7) = y · y + y · 7 + 5 · y + 5 · 7 = y2 + 7y + 5y + 36 = y2 + 12y + 36 Ex. 2 Find (3x - 5)(5x + 2) (3x - 5)(5x + 2) = 3x · 5x + 3x · 2 + -5 · 5x + -5 · 2 = 15x2 + 6x – 25x - 10 = 15x2 - 19y - 10

Ex. 3 Find (2x - 5)(3x2 -5x + 4) Ex. 4 Find (x2 – 5x + 4)(2x2 + x – 7) (2x - 5)(3x2 -5x + 4) = 2x(3x2 -5x + 4) – 5(3x2 -5x + 4) = 6x3 - 10x2 + 8x - 15x2 +25x -20 = 6x3 – (10 + 15)x2 + (8+25)x -20 = 6x3 – 25x2 + 33x -20 Ex. 4 Find (x2 – 5x + 4)(2x2 + x – 7) (x2 – 5x + 4)(2x2 + x – 7) = x2(2x2 + x - 7) -5x(2x2 + x – 7) +4(2x2 + x – 7) = 2x4 + x3 – 7x2 – 10x3 - 5x2 + 35x + 8x2 + 4x – 28 = 2x4 + (1 – 10)x3 + (– 7 – 5 + 8)x2 + (35 + 4)x – 28 = 2x4 – 9x3 – 4x2 + 39x – 28

x3 + 0x2 + 5x - 6 (x) 2x - 9 -9x3 - 0x2 - 45x + 54 2x4 0x3 10x2 - 12x Polynomials can also be multiplied in column form. Be careful to align like terms. x3 + 0x2 + 5x - 6 (x) 2x - 9 -9x3 - 0x2 - 45x + 54 2x4 0x3 10x2 - 12x - 9x3 - 57x