Find the Remainder (x^4-x^3+x^2-4x-1)/(x+2)

$\frac{x^{4} - x^{3} + x^{2} - 4 x - 1}{x + 2}$

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 $0$ .

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

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

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

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

				$x^{3}$
$x$	+	$2$		$x^{4}$	-	$x^{3}$	+	$x^{2}$	-	$4 x$	-	$1$
			-	$x^{4}$	-	$2 x^{3}$

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

				$x^{3}$
$x$	+	$2$		$x^{4}$	-	$x^{3}$	+	$x^{2}$	-	$4 x$	-	$1$
			-	$x^{4}$	-	$2 x^{3}$
					-	$3 x^{3}$

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

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

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

$x^{3}$

$3 x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

Multiply the new quotient term by the divisor.

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

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

$x^{3}$

$3 x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

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

$x^{3}$

$3 x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

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

$x^{3}$

$3 x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

$4 x$

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

$4 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

$4 x$

$7 x^{2}$

$14 x$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 14 x$

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

$4 x$

$7 x^{2}$

$14 x$

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

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

$4 x$

$7 x^{2}$

$14 x$

$18 x$

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

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

$4 x$

$7 x^{2}$

$14 x$

$18 x$

$1$

Divide the highest order term in the dividend $- 18 x$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$7 x$

$18$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

$4 x$

$7 x^{2}$

$14 x$

$18 x$

$1$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$3 x^{2}$	+	$7 x$	-	$18$
$x$	+	$2$		$x^{4}$	-	$x^{3}$	+	$x^{2}$	-	$4 x$	-	$1$
			-	$x^{4}$	-	$2 x^{3}$
					-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{3}$	+	$6 x^{2}$
							+	$7 x^{2}$	-	$4 x$
							-	$7 x^{2}$	-	$14 x$
									-	$18 x$	-	$1$
									-	$18 x$	-	$36$

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x - 36$

$x^{3}$

$3 x^{2}$

$7 x$

$18$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

$4 x$

$7 x^{2}$

$14 x$

$18 x$

$1$

$18 x$

$36$

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

$x^{3}$

$3 x^{2}$

$7 x$

$18$

$x$

$2$

$x^{4}$

$x^{3}$

$x^{2}$

$4 x$

$1$

$x^{4}$

$2 x^{3}$

$3 x^{3}$

$x^{2}$

$3 x^{3}$

$6 x^{2}$

$7 x^{2}$

$4 x$

$7 x^{2}$

$14 x$

$18 x$

$1$

$18 x$

$36$

$35$

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

$x^{3} - 3 x^{2} + 7 x - 18 + \frac{35}{x + 2}$

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

$35$

Find the Remainder (x^4-x^3+x^2-4x-1)/(x+2)

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