# How do you factor 3x^4 - x^3 - 9x^2 + 159x - 52?

May 29, 2015

Warning: the following might be (is likely to be) an incomplete solution.

Graphing the $y =$ the given expression
That is $y = 3 {x}^{4} - {x}^{3} - 9 {x}^{2} + 159 x - 52$
(see below)
suggests $x = - 4$ might be a zero for the given expression.

That is $\left(x + 4\right)$ might be a factor of $3 {x}^{4} - {x}^{3} - 9 {x}^{2} + 159 x - 52$

Using synthetic division confirms this possibility and gives
$3 {x}^{4} - {x}^{3} - 9 {x}^{2} + 159 x - 52 = \left(x + 4\right) \left(3 {x}^{3} - 13 {x}^{2} + 43 x - 13\right)$
as a factorization.

So far I have not been able to factor this second term. Perhaps someone else may do better.
graph{3x^4-x^3-9x^2+159x -52 [-10, 10, -5.21, 5.21]}