Question

use the long division method to find the result when $12x^3 - 29x^2 + 6x + 14$ is divided by $4x - 7$.

Explanation:

Step1: Divide leading terms

$\frac{12x^3}{4x} = 3x^2$
Multiply divisor by $3x^2$: $3x^2(4x-7)=12x^3-21x^2$
Subtract from dividend:
$(12x^3-29x^2+6x+14)-(12x^3-21x^2) = -8x^2+6x+14$

Step2: Divide new leading terms

$\frac{-8x^2}{4x} = -2x$
Multiply divisor by $-2x$: $-2x(4x-7)=-8x^2+14x$
Subtract from current polynomial:
$(-8x^2+6x+14)-(-8x^2+14x) = -8x+14$

Step3: Divide new leading terms

$\frac{-8x}{4x} = -2$
Multiply divisor by $-2$: $-2(4x-7)=-8x+14$
Subtract from current polynomial:
$(-8x+14)-(-8x+14) = 0$

Answer:

$3x^2 - 2x - 2$