Find the Remainder 3x^4-7x^3+5x^2-11x-2 ; x+1

$3 x^{4} - 7 x^{3} + 5 x^{2} - 11 x - 2$ ; $x + 1$

Step 1

Divide the higher order polynomial by the other polynomial in order to find the remainder.

$\frac{3 x^{4} - 7 x^{3} + 5 x^{2} - 11 x - 2}{x + 1}$

Step 2

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

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

Step 3

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

Step 4

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

Step 5

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

				$3 x^{3}$
$x$	+	$1$		$3 x^{4}$	-	$7 x^{3}$	+	$5 x^{2}$	-	$11 x$	-	$2$
			-	$3 x^{4}$	-	$3 x^{3}$

Step 6

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

				$3 x^{3}$
$x$	+	$1$		$3 x^{4}$	-	$7 x^{3}$	+	$5 x^{2}$	-	$11 x$	-	$2$
			-	$3 x^{4}$	-	$3 x^{3}$
					-	$10 x^{3}$

Step 7

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

Step 8

Divide the highest order term in the dividend $- 10 x^{3}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$10 x^{2}$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

Step 9

Multiply the new quotient term by the divisor.

				$3 x^{3}$	-	$10 x^{2}$
$x$	+	$1$		$3 x^{4}$	-	$7 x^{3}$	+	$5 x^{2}$	-	$11 x$	-	$2$
			-	$3 x^{4}$	-	$3 x^{3}$
					-	$10 x^{3}$	+	$5 x^{2}$
					-	$10 x^{3}$	-	$10 x^{2}$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{3} - 10 x^{2}$

$3 x^{3}$

$10 x^{2}$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

Step 11

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

$3 x^{3}$

$10 x^{2}$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

Step 12

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

$3 x^{3}$

$10 x^{2}$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

$11 x$

Step 13

Divide the highest order term in the dividend $15 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$10 x^{2}$

$15 x$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

$11 x$

Step 14

Multiply the new quotient term by the divisor.

$3 x^{3}$

$10 x^{2}$

$15 x$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

$11 x$

$15 x^{2}$

$15 x$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $15 x^{2} + 15 x$

$3 x^{3}$

$10 x^{2}$

$15 x$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

$11 x$

$15 x^{2}$

$15 x$

Step 16

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

$3 x^{3}$

$10 x^{2}$

$15 x$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

$11 x$

$15 x^{2}$

$15 x$

$26 x$

Step 17

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

$3 x^{3}$

$10 x^{2}$

$15 x$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

$11 x$

$15 x^{2}$

$15 x$

$26 x$

$2$

Step 18

Divide the highest order term in the dividend $- 26 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

$10 x^{2}$

$15 x$

$26$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

$11 x$

$15 x^{2}$

$15 x$

$26 x$

$2$

Step 19

Multiply the new quotient term by the divisor.

				$3 x^{3}$	-	$10 x^{2}$	+	$15 x$	-	$26$
$x$	+	$1$		$3 x^{4}$	-	$7 x^{3}$	+	$5 x^{2}$	-	$11 x$	-	$2$
			-	$3 x^{4}$	-	$3 x^{3}$
					-	$10 x^{3}$	+	$5 x^{2}$
					+	$10 x^{3}$	+	$10 x^{2}$
							+	$15 x^{2}$	-	$11 x$
							-	$15 x^{2}$	-	$15 x$
									-	$26 x$	-	$2$
									-	$26 x$	-	$26$

Step 20

The expression needs to be subtracted from the dividend, so change all the signs in $- 26 x - 26$

$3 x^{3}$

$10 x^{2}$

$15 x$

$26$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

$11 x$

$15 x^{2}$

$15 x$

$26 x$

$2$

$26 x$

$26$

Step 21

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

$3 x^{3}$

$10 x^{2}$

$15 x$

$26$

$x$

$1$

$3 x^{4}$

$7 x^{3}$

$5 x^{2}$

$11 x$

$2$

$3 x^{4}$

$3 x^{3}$

$10 x^{3}$

$5 x^{2}$

$10 x^{3}$

$10 x^{2}$

$15 x^{2}$

$11 x$

$15 x^{2}$

$15 x$

$26 x$

$2$

$26 x$

$26$

$24$

Step 22

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

$3 x^{3} - 10 x^{2} + 15 x - 26 + \frac{24}{x + 1}$

Step 23

The remainder is the part of the answer that is left after the division by $x + 1$ is complete.

$24$

Find the Remainder 3x^4-7x^3+5x^2-11x-2 ; x+1

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