Math CentralQuandaries & Queries


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,


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