Divide Using Long Polynomial Division Use the long division method to find the result when 12x^3+23x^2+30x+25 is divided by 4x+5

Use the long division method to find the result when $12 x^{3} + 23 x^{2} + 30 x + 25$ is divided by $4 x + 5$

Step 1

Write the problem as a mathematical expression.

Use the long division method to find the result when $\frac{12 x^{3} + 23 x^{2} + 30 x + 25}{4 x + 5}$

Step 2

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

$4 x$

$5$

$12 x^{3}$

$23 x^{2}$

$30 x$

$25$

Step 3

Divide the highest order term in the dividend $12 x^{3}$ by the highest order term in divisor $4 x$ .

					$3 x^{2}$
	$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$

Step 4

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			+	$12 x^{3}$	+	$15 x^{2}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{3} + 15 x^{2}$

				$3 x^{2}$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$

Step 6

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

				$3 x^{2}$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$

Step 7

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

				$3 x^{2}$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$

Step 8

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $4 x$ .

				$3 x^{2}$	+	$2 x$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$

Step 9

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$2 x$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$
					+	$8 x^{2}$	+	$10 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} + 10 x$

				$3 x^{2}$	+	$2 x$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$
					-	$8 x^{2}$	-	$10 x$

Step 11

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

				$3 x^{2}$	+	$2 x$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$
					-	$8 x^{2}$	-	$10 x$
							+	$20 x$

Step 12

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

				$3 x^{2}$	+	$2 x$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$
					-	$8 x^{2}$	-	$10 x$
							+	$20 x$	+	$25$

Step 13

Divide the highest order term in the dividend $20 x$ by the highest order term in divisor $4 x$ .

				$3 x^{2}$	+	$2 x$	+	$5$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$
					-	$8 x^{2}$	-	$10 x$
							+	$20 x$	+	$25$

Step 14

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$2 x$	+	$5$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$
					-	$8 x^{2}$	-	$10 x$
							+	$20 x$	+	$25$
							+	$20 x$	+	$25$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $20 x + 25$

				$3 x^{2}$	+	$2 x$	+	$5$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$
					-	$8 x^{2}$	-	$10 x$
							+	$20 x$	+	$25$
							-	$20 x$	-	$25$

Step 16

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

				$3 x^{2}$	+	$2 x$	+	$5$
$4 x$	+	$5$		$12 x^{3}$	+	$23 x^{2}$	+	$30 x$	+	$25$
			-	$12 x^{3}$	-	$15 x^{2}$
					+	$8 x^{2}$	+	$30 x$
					-	$8 x^{2}$	-	$10 x$
							+	$20 x$	+	$25$
							-	$20 x$	-	$25$
										$0$

Step 17

Since the remander is $0$ , the final answer is the quotient.

$3 x^{2} + 2 x + 5$

Divide Using Long Polynomial Division Use the long division method to find the result when 12x^3+23x^2+30x+25 is divided by 4x+5

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