Question

one factor of this polynomial is (2x - 1).
$2x^3 + 5x^2 - 27x + 12$
use long division to determine which expression represents the other factor.
$\circ$ $x^2 + 3x - 12$
$\circ$ $x^2 + 12x - 3$
$\circ$ $x^2 - 3x + 12$
$\circ$ $x^2 - 12x + 3$

Explanation:

Step1: Divide leading terms

$\frac{2x^3}{2x} = x^2$
Multiply divisor by $x^2$: $(2x-1)x^2 = 2x^3 - x^2$
Subtract from dividend: $(2x^3+5x^2-27x+12)-(2x^3-x^2) = 6x^2-27x+12$

Step2: Divide new leading terms

$\frac{6x^2}{2x} = 3x$
Multiply divisor by $3x$: $(2x-1)3x = 6x^2 - 3x$
Subtract: $(6x^2-27x+12)-(6x^2-3x) = -24x+12$

Step3: Divide final leading terms

$\frac{-24x}{2x} = -12$
Multiply divisor by $-12$: $(2x-1)(-12) = -24x+12$
Subtract: $(-24x+12)-(-24x+12) = 0$

Answer:

A. $x^2 + 3x - 12$