Divide Using Long Polynomial Division (x^4+5x^3+6x^2-x-2)÷(x+2)

$(x^{4} + 5 x^{3} + 6 x^{2} - x - 2) \div (x + 2)$

Step 1

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

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

Step 2

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

$x^{3}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

Step 4

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

$x^{3}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

Step 5

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

$x^{3}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

Step 6

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

$x^{3}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$6 x^{2}$

Step 7

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

$x^{3}$

$3 x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$6 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$6 x^{2}$

$3 x^{3}$

$6 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} + 6 x^{2}$

$x^{3}$

$3 x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$6 x^{2}$

$3 x^{3}$

$6 x^{2}$

Step 10

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

$x^{3}$

$3 x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$6 x^{2}$

$3 x^{3}$

$6 x^{2}$

$0$

Step 11

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

$x^{3}$

$3 x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$6 x^{2}$

$3 x^{3}$

$6 x^{2}$

$0$

$x$

$2$

Step 12

Divide the highest order term in the dividend $- x$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$0 x$

$1$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$6 x^{2}$

$3 x^{3}$

$6 x^{2}$

$0$

$x$

$2$

Step 13

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
$x$	+	$2$		$x^{4}$	+	$5 x^{3}$	+	$6 x^{2}$	-	$x$	-	$2$
			-	$x^{4}$	-	$2 x^{3}$
					+	$3 x^{3}$	+	$6 x^{2}$
					-	$3 x^{3}$	-	$6 x^{2}$
								$0$	-	$x$	-	$2$
									-	$x$	-	$2$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- x - 2$

$x^{3}$

$3 x^{2}$

$0 x$

$1$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$6 x^{2}$

$3 x^{3}$

$6 x^{2}$

$0$

$x$

$2$

$x$

$2$

Step 15

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

$x^{3}$

$3 x^{2}$

$0 x$

$1$

$x$

$2$

$x^{4}$

$5 x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$6 x^{2}$

$3 x^{3}$

$6 x^{2}$

$0$

$x$

$2$

$x$

$2$

$0$

Step 16

Since the remander is $0$ , the final answer is the quotient.

$x^{3} + 3 x^{2} + 0 x - 1$

Divide Using Long Polynomial Division (x^4+5x^3+6x^2-x-2)÷(x+2)

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