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6.4 Factoring Trinomials and

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Let’s Investigate: Let’s Investigate: (x +4)(x + 3 ) = x 2 +3x +4x +12 = x 2 + 7x +21

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(x +4)(x + 3) = x 2 +3x +4x +12 = x 2 + 7x +12 Factor: x 2 +7x +12 What set of factors of “12” add up to “7”

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The Diamond Method product of “a” and “c” “b” Find the two numbers that will multiply to get the top and add to get the bottom.

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(x +4)(x + 3) = x 2 +3x +4x +12 = x 2 + 7x +12 Factor: x 2 +7x +12 What set of factors of “12” add up to “7” (using the diamond method) 12 7 34

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Factoring by grouping using the diamond method Use the numbers in the left and right of your diamond as your “x” coefficients to make a 4-term polynomial x 2 + 3x + 4x + 12 Group the first two terms together and the last two terms together (x 2 + 3x) + (4x + 12) Factor out the GCF of each group x(x + 3) + 4(x + 3) Notice that the contents of the parentheses are the same! Use that group as one set of parentheses, and then take the GCF's and put them in a set of parentheses (x + 3)(x + 4) Voila! You've factored the polynomial!

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Factoring the “bottoms-up” way using the diamond method Set up two sets of parentheses as shown below using the left and right numbers in the diamond (__x + 3)(__x + 4) Divide both of the “numbers” by “a” (__x + 3)(__x + 4) 1 1 “Bottoms-up!” Simplify... (x + 3)(x + 4) Done!

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Factor using your favorite method:

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Now...let's try one where the “a” isn't “1” 36 13 94 Let's try it by grouping, first: 6x 2 + 9x + 4x + 6 (6x 2 + 9x) + (4x + 6) 3x(2x + 3) + 2(2x + 3) (2x + 3)(3x + 2) done!

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Now...the “bottoms-up” method Same equation...6x 2 + 13x + 6 (__ x+ 9)(__x +4) 6 6 Reduce... (__ x+ 3)(__x +2) 2 3 Bottoms-up! (2x + 3)(3x + 2) Done!

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Factor using your favorite method: 3x 2 + 4x + 1

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Factor using your favorite method: 8x 2 - 6x - 9

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A#: Page 276 # 1 – 36 (show the diamond method)

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Rational Expressions: Multiply & Divide Polynomials Aim: Simplify a polynomial expression by multiplying and/or dividing polynomials.

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