   SEARCH HOME Math Central Quandaries & 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,

Leeanne     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.