Question

question
use the long division method to find the result when ( 3x^3 + 5x^2 - 24x + 25 ) is divided by ( 3x - 4 ). if there is a remainder, express the result in the form ( q(x) + \frac{r(x)}{b(x)} ).

Explanation:

Step1: Divide leading terms

$\frac{3x^3}{3x} = x^2$
Multiply divisor by $x^2$: $x^2(3x-4)=3x^3-4x^2$
Subtract from dividend:
$(3x^3+5x^2-24x+25)-(3x^3-4x^2)=9x^2-24x+25$

Step2: Divide new leading terms

$\frac{9x^2}{3x}=3x$
Multiply divisor by $3x$: $3x(3x-4)=9x^2-12x$
Subtract from current polynomial:
$(9x^2-24x+25)-(9x^2-12x)=-12x+25$

Step3: Divide next leading terms

$\frac{-12x}{3x}=-4$
Multiply divisor by $-4$: $-4(3x-4)=-12x+16$
Subtract from current polynomial:
$(-12x+25)-(-12x+16)=9$

Answer:

$x^2 + 3x - 4 + \frac{9}{3x-4}$