Expert: Scott A Wilson - 7/13/2010

Question
When grouping polynomials (trinomials) into binomials I undestand the process but I spend way to much time placing things in the terms in their proper order and doing checks, mainly with negative signs. Is there a short cut I can zero on placement after I decide which factors to use to equal the sum of the middle term? For example if I have... 2(xto2ndpwr) + 13x - 7 = ( x + )( x - ) how can I do this without blinking?  I really have problems at this point. Is there a short cut?

Answer
I have found no way to do it simply.
For this problem, 2x² + 13x - 7, know the factors of the 1st and last terms.
Since both 2 and 7 are prime, and 2*7 is 14, the 2 and 7 go in opposite parenthesis.

That is, it is (2x 1)(x 7).  The signs need to be opposite, and we need a 14 - 1 to get to 13
(as opposed to a 1 - 14), so take (2x-1)(x+7).

I use the quadratic formula to find the factors if factoring is not obvious at first.
The quadratic formula is [-b ± √(b²-4ac)]/(2a).  13²=169 and 4*2*7=56.
The solutions here would be (-13±√(169+56))/4, which is (-13±√225)/4.
It is known that √225 = 15, so we have (-13±15)/4, which gives 2/4 and -28/4.
Now 2/4 = 1/2 and -28/4=-7, so the factors are (x - 1/2)(x+7).
This gives x² + 13x/2 - 7/2, and times 2 gives 2x² + 13x - 7.

After doing so much math for over 20 years, I know the factors of the first 100 digits and many more and can do it by trial and error withou too much, but it still takes a moment or two sometimes.



For example, if we had 12x² +64x - 48, notice that all terms are divisible by 4, so that we have
4(3x²+16x-12).

Now 3 is prime, so we have 1 and 3 as the leading coefficients.  The number 12, however, is 1*12, 2*6, or 3*4.

Now if we take the 1 and 3 along with the 1 and 12, it would be 12*1 - 1*3 = 9, and that's not 16, so take the next.

The next is 2 and 6, and 3*6 - 2*1 = 16, so that's the choice to use.
That is, 1 and 2 on the outside and 3 and 6 on the inside.  This would be (x+6)(3x-2).
When this has been gotten, remember that the whole expression has a 4 out front,
so it is 4(x+6)(3x-2).