Problem Statement
Using remainder and factor theorem, factorize completely: $2x^3 + x^2 - 13x + 6$.
Verified Solution & Marking Scheme
Find First Factor
Let $f(x) = 2x^3 + x^2 - 13x + 6$.
$f(2) = 2(8) + 4 - 26 + 6 = 16 + 4 - 26 + 6 = 0$. Hence $(x - 2)$ is a factor.
Divide Polynomial
$2x^3 + x^2 - 13x + 6 = (x - 2)(2x^2 + 5x - 3)$
Factorize Quadratic
$2x^2 + 5x - 3 = (2x - 1)(x + 3)$