# How do you factor completely 5x^3 - 26x^2 + 5x?

May 12, 2016

x(5x - 1)(x - 5)

#### Explanation:

$f \left(x\right) = x y = x \left(5 {x}^{2} - 26 x + 5\right)$
Factor the trinomial y by the new AC Method (Socratic Search)
$y = 5 {x}^{2} - 26 x + 5 = 5 \left(x + p\right) \left(x + q\right)$
Converted trinomial: $y ' = {x}^{2} - 26 x + 25 =$ (x + p')(x + q')
Since a + b + c = 0, then p' = -1 and q' = -25.
Back to y, $p = \frac{p '}{a} = - \frac{1}{5}$ and $q = \frac{q '}{a} = - \frac{25}{5} = - 5.$
Factored form $y = 5 \left(x - \frac{1}{5}\right) \left(x - 5\right) = \left(5 x - 1\right) \left(x - 5\right)$
f(x) = x(5x - 1)(x - 5)