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6.3 Adding, Subtracting, & Multiplying Polynomials p. 338

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To + or -, + or – the coeff. of like terms! Vertical format : Add 3x 3 +2x 2 -x-7 and x 3 -10x 2 +8. 3x 3 + 2x 2 – x – 7 + x 3 – 10x 2 + 8 Line up like terms 4x 3 – 8x 2 – x + 1

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Horizontal format : Combine like terms (8x 3 – 3x 2 – 2x + 9) – (2x 3 + 6x 2 – x + 1)= (8x 3 – 2x 3 )+(-3x 2 – 6x 2 )+(-2x + x) + (9 – 1)= 6x 3 + -9x 2 + -x + 8 = 6x 3 – 9x 2 – x + 8

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Examples: Adding & Subtracting (9x 3 – 2x + 1) + (5x 2 + 12x -4) = 9x 3 + 5x 2 – 2x + 12x + 1 – 4 = 9x 3 + 5x 2 + 10x – 3 (2x 2 + 3x) – (3x 2 + x – 4)= 2x 2 + 3x – 3x 2 – x + 4 = 2x 2 - 3x 2 + 3x – x + 4 = -x 2 + 2x + 4

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Multiplying Polynomials: Vertically (-x 2 + 2x + 4)(x – 3)= -x 2 + 2x + 4 * x – 3 3x 2 – 6x – 12 -x 3 + 2x 2 + 4x -x 3 + 5x 2 – 2x – 12

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Multiplying Polynomials : Horizontally (x – 3)(3x 2 – 2x – 4)= (x – 3)(3x 2 ) + (x – 3)(-2x) + (x – 3)(-4) = (3x 3 – 9x 2 ) + (-2x 2 + 6x) + (-4x + 12) = 3x 3 – 9x 2 – 2x 2 + 6x – 4x +12 = 3x 3 – 11x 2 + 2x + 12

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Multiplying 3 Binomials : (x – 1)(x + 4)(x + 3) = FOIL the first two: (x 2 – x +4x – 4)(x + 3) = (x 2 + 3x – 4)(x + 3) = Then multiply the trinomial by the binomial (x 2 + 3x – 4)(x) + (x 2 + 3x – 4)(3) = (x 3 + 3x 2 – 4x) + (3x 2 + 9x – 12) = x 3 + 6x 2 + 5x - 12

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Some binomial products appear so much we need to recognize the patterns! Sum & Difference (S&D): (a + b)(a – b) = a 2 – b 2 Example: (x + 3)(x – 3) = x 2 – 9 Square of Binomial: (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2

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Last Pattern Cube of a Binomial (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (a – b) 3 = a 3 - 3a 2 b + 3ab 2 – b 3

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Example: (x + 5) 3 = a = x and b = 5 x 3 + 3(x) 2 (5) + 3(x)(5) 2 + (5) 3 = x 3 + 15x 2 + 75x + 125

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Assignment

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