How do you factor x^2-26x^3-27?

1 Answer
Feb 4, 2015

Factoring a polynomial (when possible using real numbers) means to find its roots, and then divide the polynomial for the linear factor $\left(x - {x}_{i}\right)$, where ${x}_{i}$ is the root.

In your case, you can see that $x = - 1$ is a root, since
${\left(- 1\right)}^{2} - 26 {\left(- 1\right)}^{3} - 27 = 1 + 26 - 27 = 0$.

Dividing your polynomial by $\left(x + 1\right)$, you obtain $- \left(26 {x}^{2} - 27 x + 27\right)$. The discriminant of this quadratic polynomial is $- 2079$, and thus it has no real solutions. We cannot further simplify it, and so the answer is
$- \left(x + 1\right) \left(26 {x}^{2} - 27 x + 27\right)$