   SEARCH HOME Math Central Quandaries & Queries  Question from Sean, a student: How would I factor this polynomial? 2x^3 + 11x^2 - 7x - 6 Hi Sean.

Factoring cubic generally is not simple. I would suggest using the factor theorem. This tells us that given a cubic f(x), if we can guess a value "n" for which f(n) = 0, then (x - n) is a factor of f(x). Then we can use synthetic division or long division to give us a quadratic factor to go with the (x - n). Factoring a quadratic should be straightforward for you if you are working on factoring cubics.

For example:

f(x) = 4x3 - 6x2 + 8x - 6

I try guessing x = 1. Does f(1) = 0?

f(1) = 4(1)3 - 6(1)2 + 8(1) - 6 = 4 - 6 + 8 - 6 = 0. Getting it equal to zero means (x - 1) is a factor. (I got it on my first guess)

Now I use synthetic division to divide (x - 1) into 4x3 - 6x2 + 8x - 6 :

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So f(x) = (x - 1) (4x2 - 2x + 6).

(If you need to review how synthetic division works, see Melanie's explanation)

Hope this helps,
Stephen La Rocque     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.