Divide Using Long Polynomial Division (x^4-2x^3-10x^2+5x+36)/(x^2-3x-4)

$\frac{x^{4} - 2 x^{3} - 10 x^{2} + 5 x + 36}{x^{2} - 3 x - 4}$

Step 1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

Step 2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x^{2}$ .

							$x^{2}$
	$x^{2}$	-	$3 x$	-	$4$		$x^{4}$	-	$2 x^{3}$	-	$10 x^{2}$	+	$5 x$	+	$36$

Step 3

Multiply the new quotient term by the divisor.

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 3 x^{3} - 4 x^{2}$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

Step 5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

Step 6

Pull the next terms from the original dividend down into the current dividend.

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

Step 7

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

Step 8

Multiply the new quotient term by the divisor.

$x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$4 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 3 x^{2} - 4 x$

$x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$4 x$

Step 10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

Step 11

Pull the next terms from the original dividend down into the current dividend.

$x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$36$

Step 12

Divide the highest order term in the dividend $- 3 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$x$

$3$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$36$

Step 13

Multiply the new quotient term by the divisor.

$x^{2}$

$x$

$3$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$36$

$3 x^{2}$

$9 x$

$12$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{2} + 9 x + 12$

$x^{2}$

$x$

$3$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$36$

$3 x^{2}$

$9 x$

$12$

Step 15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{2}$

$x$

$3$

$x^{2}$

$3 x$

$4$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$5 x$

$36$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{3}$

$6 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$36$

$3 x^{2}$

$9 x$

$12$

$24$

Step 16

The final answer is the quotient plus the remainder over the divisor.

$x^{2} + x - 3 + \frac{24}{x^{2} - 3 x - 4}$

Divide Using Long Polynomial Division (x^4-2x^3-10x^2+5x+36)/(x^2-3x-4)

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