Divide Using Long Polynomial Division (x^3+5x^2+5x-3)÷(x^2+3x-1)

$(x^{3} + 5 x^{2} + 5 x - 3) \div (x^{2} + 3 x - 1)$

Step 1

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

$x^{2}$

$3 x$

$1$

$x^{3}$

$5 x^{2}$

$5 x$

$3$

Step 2

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

$x$

$x^{2}$

$3 x$

$1$

$x^{3}$

$5 x^{2}$

$5 x$

$3$

Step 3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$3 x$

$1$

$x^{3}$

$5 x^{2}$

$5 x$

$3$

$x^{3}$

$3 x^{2}$

$x$

Step 4

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

$x$

$x^{2}$

$3 x$

$1$

$x^{3}$

$5 x^{2}$

$5 x$

$3$

$x^{3}$

$3 x^{2}$

$x$

Step 5

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

$x$

$x^{2}$

$3 x$

$1$

$x^{3}$

$5 x^{2}$

$5 x$

$3$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$6 x$

Step 6

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

$x$

$x^{2}$

$3 x$

$1$

$x^{3}$

$5 x^{2}$

$5 x$

$3$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$6 x$

$3$

Step 7

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

$x$

$2$

$x^{2}$

$3 x$

$1$

$x^{3}$

$5 x^{2}$

$5 x$

$3$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$6 x$

$3$

Step 8

Multiply the new quotient term by the divisor.

$x$

$2$

$x^{2}$

$3 x$

$1$

$x^{3}$

$5 x^{2}$

$5 x$

$3$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$6 x$

$3$

$2 x^{2}$

$6 x$

$2$

Step 9

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

$x$

$2$

$x^{2}$

$3 x$

$1$

$x^{3}$

$5 x^{2}$

$5 x$

$3$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$6 x$

$3$

$2 x^{2}$

$6 x$

$2$

Step 10

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

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

Step 11

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

$x + 2 - \frac{1}{x^{2} + 3 x - 1}$

Divide Using Long Polynomial Division (x^3+5x^2+5x-3)÷(x^2+3x-1)

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