Find the Remainder (3x^4+5x^2-8x-10)÷(x+3)

$(3 x^{4} + 5 x^{2} - 8 x - 10) \div (x + 3)$

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 $0$ .

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

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

$3 x^{3}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} + 9 x^{3}$

$3 x^{3}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

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

$3 x^{3}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

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

$3 x^{3}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

Divide the highest order term in the dividend $- 9 x^{3}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{3} - 27 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

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

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

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

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

$8 x$

Divide the highest order term in the dividend $32 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$9 x^{2}$

$32 x$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

$8 x$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$9 x^{2}$

$32 x$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

$8 x$

$32 x^{2}$

$96 x$

The expression needs to be subtracted from the dividend, so change all the signs in $32 x^{2} + 96 x$

$3 x^{3}$

$9 x^{2}$

$32 x$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

$8 x$

$32 x^{2}$

$96 x$

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

$3 x^{3}$

$9 x^{2}$

$32 x$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

$8 x$

$32 x^{2}$

$96 x$

$104 x$

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

$3 x^{3}$

$9 x^{2}$

$32 x$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

$8 x$

$32 x^{2}$

$96 x$

$104 x$

$10$

Divide the highest order term in the dividend $- 104 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

$9 x^{2}$

$32 x$

$104$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

$8 x$

$32 x^{2}$

$96 x$

$104 x$

$10$

Multiply the new quotient term by the divisor.

				$3 x^{3}$	-	$9 x^{2}$	+	$32 x$	-	$104$
$x$	+	$3$		$3 x^{4}$	+	$0 x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$10$
			-	$3 x^{4}$	-	$9 x^{3}$
					-	$9 x^{3}$	+	$5 x^{2}$
					+	$9 x^{3}$	+	$27 x^{2}$
							+	$32 x^{2}$	-	$8 x$
							-	$32 x^{2}$	-	$96 x$
									-	$104 x$	-	$10$
									-	$104 x$	-	$312$

The expression needs to be subtracted from the dividend, so change all the signs in $- 104 x - 312$

$3 x^{3}$

$9 x^{2}$

$32 x$

$104$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

$8 x$

$32 x^{2}$

$96 x$

$104 x$

$10$

$104 x$

$312$

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

$3 x^{3}$

$9 x^{2}$

$32 x$

$104$

$x$

$3$

$3 x^{4}$

$0 x^{3}$

$5 x^{2}$

$8 x$

$10$

$3 x^{4}$

$9 x^{3}$

$5 x^{2}$

$9 x^{3}$

$27 x^{2}$

$32 x^{2}$

$8 x$

$32 x^{2}$

$96 x$

$104 x$

$10$

$104 x$

$312$

$302$

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

$3 x^{3} - 9 x^{2} + 32 x - 104 + \frac{302}{x + 3}$

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

$302$

Find the Remainder (3x^4+5x^2-8x-10)÷(x+3)

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