Divide Using Long Polynomial Division (16x^3-3x+5)÷(4x+3)

$(16 x^{3} - 3 x + 5) \div (4 x + 3)$

Step 1

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

$4 x$

$3$

$16 x^{3}$

$0 x^{2}$

$3 x$

$5$

Step 2

Divide the highest order term in the dividend $16 x^{3}$ by the highest order term in divisor $4 x$ .

					$4 x^{2}$
	$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$

Step 3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			+	$16 x^{3}$	+	$12 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{3} + 12 x^{2}$

				$4 x^{2}$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$

Step 5

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

				$4 x^{2}$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$

Step 6

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

				$4 x^{2}$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$

Step 7

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $4 x$ .

				$4 x^{2}$	-	$3 x$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$

Step 8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$3 x$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$
					-	$12 x^{2}$	-	$9 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} - 9 x$

				$4 x^{2}$	-	$3 x$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$
					+	$12 x^{2}$	+	$9 x$

Step 10

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

				$4 x^{2}$	-	$3 x$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$
					+	$12 x^{2}$	+	$9 x$
							+	$6 x$

Step 11

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

				$4 x^{2}$	-	$3 x$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$
					+	$12 x^{2}$	+	$9 x$
							+	$6 x$	+	$5$

Step 12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $4 x$ .

				$4 x^{2}$	-	$3 x$	+	$\frac{3}{2}$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$
					+	$12 x^{2}$	+	$9 x$
							+	$6 x$	+	$5$

Step 13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$3 x$	+	$\frac{3}{2}$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$
					+	$12 x^{2}$	+	$9 x$
							+	$6 x$	+	$5$
							+	$6 x$	+	$\frac{9}{2}$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + \frac{9}{2}$

				$4 x^{2}$	-	$3 x$	+	$\frac{3}{2}$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$
					+	$12 x^{2}$	+	$9 x$
							+	$6 x$	+	$5$
							-	$6 x$	-	$\frac{9}{2}$

Step 15

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

				$4 x^{2}$	-	$3 x$	+	$\frac{3}{2}$
$4 x$	+	$3$		$16 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$5$
			-	$16 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	-	$3 x$
					+	$12 x^{2}$	+	$9 x$
							+	$6 x$	+	$5$
							-	$6 x$	-	$\frac{9}{2}$
									+	$\frac{1}{2}$

Step 16

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

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

Divide Using Long Polynomial Division (16x^3-3x+5)÷(4x+3)

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