Find the Remainder (x^4+x^3-3x-3)÷(x+2)

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

Step 1

To calculate the remainder, first divide the polynomials.

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

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

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

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

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

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

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

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

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

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$3 x$

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

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$3 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$4 x$

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

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$4 x$

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

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$4 x$

$7 x$

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

$x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$4 x$

$7 x$

$3$

Divide the highest order term in the dividend $- 7 x$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$2 x$

$7$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$4 x$

$7 x$

$3$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$x^{2}$	+	$2 x$	-	$7$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$3$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	+	$0 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							+	$2 x^{2}$	-	$3 x$
							-	$2 x^{2}$	-	$4 x$
									-	$7 x$	-	$3$
									-	$7 x$	-	$14$

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x - 14$

$x^{3}$

$x^{2}$

$2 x$

$7$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$4 x$

$7 x$

$3$

$7 x$

$14$

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

$x^{3}$

$x^{2}$

$2 x$

$7$

$x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$3 x$

$3$

$x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$4 x$

$7 x$

$3$

$7 x$

$14$

$11$

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

$x^{3} - x^{2} + 2 x - 7 + \frac{11}{x + 2}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$11$

Find the Remainder (x^4+x^3-3x-3)÷(x+2)

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