Question

use the long division method to find the result when $8x^4 + 18x^3 - 22x^2 - 30x + 24$ is divided by $2x^2 + 2x - 5$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Divide leading terms

$\frac{8x^4}{2x^2} = 4x^2$

Step2: Multiply divisor by $4x^2$

$4x^2(2x^2 + 2x - 5) = 8x^4 + 8x^3 - 20x^2$

Step3: Subtract from dividend

$(8x^4 + 18x^3 - 22x^2 - 30x + 24) - (8x^4 + 8x^3 - 20x^2) = 10x^3 - 2x^2 - 30x + 24$

Step4: Divide new leading terms

$\frac{10x^3}{2x^2} = 5x$

Step5: Multiply divisor by $5x$

$5x(2x^2 + 2x - 5) = 10x^3 + 10x^2 - 25x$

Step6: Subtract from new polynomial

$(10x^3 - 2x^2 - 30x + 24) - (10x^3 + 10x^2 - 25x) = -12x^2 - 5x + 24$

Step7: Divide new leading terms

$\frac{-12x^2}{2x^2} = -1$

Step8: Multiply divisor by $-1$

$-1(2x^2 + 2x - 5) = -2x^2 - 2x + 5$

Step9: Subtract to find remainder

$(-12x^2 - 5x + 24) - (-2x^2 - 2x + 5) = -10x^2 - 3x + 19$
Wait, correction to Step9:

Step9 (corrected): Subtract to find remainder

$(-12x^2 - 5x + 24) - (-2x^2 - 2x + 5) = -10x^2 - 3x + 19$ → No, re-calculate:
$(-12x^2 + 2x^2) + (-5x + 2x) + (24 - 5) = -10x^2 - 3x + 19$
Wait, error in Step7: $\frac{-12x^2}{2x^2} = -6$, not -1. Corrected steps from Step7:

Step7 (corrected): Divide new leading terms

$\frac{-12x^2}{2x^2} = -6$

Step8 (corrected): Multiply divisor by $-6$

$-6(2x^2 + 2x - 5) = -12x^2 - 12x + 30$

Step9 (corrected): Subtract to find remainder

$(-12x^2 - 5x + 24) - (-12x^2 - 12x + 30) = 7x - 6$
Wait, final corrected steps:

Step1: Divide leading terms

$\frac{8x^4}{2x^2} = 4x^2$

Step2: Multiply divisor by $4x^2$

$4x^2(2x^2 + 2x - 5) = 8x^4 + 8x^3 - 20x^2$

Step3: Subtract from dividend

$(8x^4 + 18x^3 - 22x^2 - 30x + 24) - (8x^4 + 8x^3 - 20x^2) = 10x^3 - 2x^2 - 30x + 24$

Step4: Divide new leading terms

$\frac{10x^3}{2x^2} = 5x$

Step5: Multiply divisor by $5x$

$5x(2x^2 + 2x - 5) = 10x^3 + 10x^2 - 25x$

Step6: Subtract from new polynomial

$(10x^3 - 2x^2 - 30x + 24) - (10x^3 + 10x^2 - 25x) = -12x^2 - 5x + 24$

Step7: Divide new leading terms

$\frac{-12x^2}{2x^2} = -6$

Step8: Multiply divisor by $-6$

$-6(2x^2 + 2x - 5) = -12x^2 - 12x + 30$

Step9: Subtract to find remainder

$(-12x^2 - 5x + 24) - (-12x^2 - 12x + 30) = 7x - 6$

Step10: Combine quotient and remainder

$q(x) + \frac{r(x)}{b(x)} = 4x^2 + 5x - 6 + \frac{7x - 6}{2x^2 + 2x - 5}$

Final correct answer:

Step1: Divide leading terms

$\frac{8x^4}{2x^2} = 4x^2$

Step2: Multiply divisor by $4x^2$

$4x^2(2x^2 + 2x - 5) = 8x^4 + 8x^3 - 20x^2$

Step3: Subtract from dividend

$(8x^4 + 18x^3 - 22x^2 - 30x + 24) - (8x^4 + 8x^3 - 20x^2) = 10x^3 - 2x^2 - 30x + 24$

Step4: Divide new leading terms

$\frac{10x^3}{2x^2} = 5x$

Step5: Multiply divisor by $5x$

$5x(2x^2 + 2x - 5) = 10x^3 + 10x^2 - 25x$

Step6: Subtract from new polynomial

$(10x^3 - 2x^2 - 30x + 24) - (10x^3 + 10x^2 - 25x) = -12x^2 - 5x + 24$

Step7: Divide new leading terms

$\frac{-12x^2}{2x^2} = -6$

Step8: Multiply divisor by $-6$

$-6(2x^2 + 2x - 5) = -12x^2 - 12x + 30$

Step9: Subtract to find remainder

$(-12x^2 - 5x + 24) - (-12x^2 - 12x + 30) = 7x - 6$

Step10: Form final expression

$q(x) + \frac{r(x)}{b(x)} = 4x^2 + 5x - 6 + \frac{7x - 6}{2x^2 + 2x - 5}$

Answer:

$4x^2 + 5x - 1 + \frac{-5x + 19}{2x^2 + 2x - 5}$