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| Math Central | Quandaries & Queries |
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Question from Kim, a student: Hi, could you please help me solve this. 3x(squared) +20x - 7 |
A trinomial of this type may be factored by decomposition.
The initial goal is to break down (decompose) the middle term of +20x into two terms so that we can proceed to factor by grouping the resulting 4 terms. In order to properly break down the +20x we need to examine the coefficient on x2 and the constant. If we multiply these numbers we get (3)(-7) = -21. We must find two numbers whose:
(1) product is -21 and
(2) sum is +20 (from the middle term)
The two numbers we want are +21 and -1. Proceed as follows:
| 3x2 + 20x - 7 = 3x2 + 21x - 1x - 7 | break down middle term |
| = (3x2 + 21x) +(-1x - 7) | group terms in twos (be careful with negatives!) |
| = 3x(x+7) - 1 (x+7) | factor out a gcf from each group |
| = (x+7)(3x-1) | the two factors of the polynomial are formed by: |
| (a) the common factor that you see: (x+7) | |
| (b) the multipliers of 3x and -1: (3x-1) |
This method of decomposition is useful for factoring a trinomial when the x2 term of the trinomial is not 1.
Hope this helps,
Leeanne
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