6 4 Factoring Trinomials and Lets Investigate x

6. 4 Factoring Trinomials and

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

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

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

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

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 + 3 x + 4 x + 12 Group the first two terms together and the last two terms together (x 2 + 3 x) + (4 x + 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!

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!

Factor using your favorite method:

Factor using your favorite method:

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

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

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

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

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