The following method for factoring ax2+bx+c, where a /= 0 is frequently called the "Master Product Method." Some books refer to it as "Factoring a Trinomial by Grouping."
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To factor ax2+bx+c, where a /= 0, |
Example: Factor 3x2-16x-12 |
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Step 1: Determine the Master Product, ac. |
Step 1: The Master Product is (3)(-12)=-36 |
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Step 2: Determine pairs of factors of the Master Product that add up to the x coefficient, b. |
Step 2: Determine pairs of factors of -36 that add up to -16. Pairs of factors of -36 are:
The pair that adds up to the x coefficient, -16, is (2)(-18). |
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Step 3: Use the two numbers determined in Step 2 to write bx as two terms. |
Step 3: In the trinomial we are factoring write -16x as 2x-18x, so we now have 3x2+2x-18x-12 |
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Step 4: Factor the resulting four terms by grouping. |
Step 4: x(3x+2)-6(3x+2) |
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Summary of This Example: 3x2-16x-12 -- Trinomial to be factored 3x2+2x-18x-12 -- Steps 2 and 3 x(3x+2)-6(3x+2) -- Step 4 (3x+2)(x-6) -- Step 4 |
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Another Example: Factor 12x2+8x-15 Because 12x2+8x-15 is a trinomial with a leading coefficient other than one, we know to use the Master Product Method. Step 1. Determine the Master Product: (12)(-15) = -180 Step 2. Find factors of -180 that add up to the x coefficient, 8, in the given trinomial. We know that the factors must be opposite in sign because their product is to be negative, -180. And we know that the larger of the factors must be the one that is positive because their sum is to be positive, +8. The possibilities are 180, -1 Of these, 18 and -10 are the factors of -180 that add up to 8. Step 3. Use the two numbers determined in Step 2 to write the middle term of the given trinomial as two terms. So we get 12x2+18x-10x-15 Step 4. Factor the resulting four terms by grouping. 6x(2x+3) - 5(2x+3) (2x+3)(6x-5) So 12x2+8x-15 = (2x+3)(6x-5). |