Algebra Examples

Use the long division method to find the result when $4 x^{3} + 23 x^{2} + 21 x + 30$ is divided by $x + 5$

Step 1

Write the problem as a mathematical expression.

Use the long division method to find the result when $\frac{4 x^{3} + 23 x^{2} + 21 x + 30}{x + 5}$

Step 2

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

$x$

$5$

$4 x^{3}$

$23 x^{2}$

$21 x$

$30$

Step 3

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

					$4 x^{2}$
	$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$

Step 4

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			+	$4 x^{3}$	+	$20 x^{2}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 20 x^{2}$

				$4 x^{2}$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$

Step 6

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

				$4 x^{2}$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$

Step 7

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

				$4 x^{2}$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$

Step 8

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

				$4 x^{2}$	+	$3 x$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$

Step 9

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$3 x$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$
					+	$3 x^{2}$	+	$15 x$

Step 10

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

				$4 x^{2}$	+	$3 x$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$
					-	$3 x^{2}$	-	$15 x$

Step 11

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

				$4 x^{2}$	+	$3 x$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$
					-	$3 x^{2}$	-	$15 x$
							+	$6 x$

Step 12

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

				$4 x^{2}$	+	$3 x$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$
					-	$3 x^{2}$	-	$15 x$
							+	$6 x$	+	$30$

Step 13

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

				$4 x^{2}$	+	$3 x$	+	$6$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$
					-	$3 x^{2}$	-	$15 x$
							+	$6 x$	+	$30$

Step 14

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$3 x$	+	$6$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$
					-	$3 x^{2}$	-	$15 x$
							+	$6 x$	+	$30$
							+	$6 x$	+	$30$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + 30$

				$4 x^{2}$	+	$3 x$	+	$6$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$
					-	$3 x^{2}$	-	$15 x$
							+	$6 x$	+	$30$
							-	$6 x$	-	$30$

Step 16

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

				$4 x^{2}$	+	$3 x$	+	$6$
$x$	+	$5$		$4 x^{3}$	+	$23 x^{2}$	+	$21 x$	+	$30$
			-	$4 x^{3}$	-	$20 x^{2}$
					+	$3 x^{2}$	+	$21 x$
					-	$3 x^{2}$	-	$15 x$
							+	$6 x$	+	$30$
							-	$6 x$	-	$30$
										$0$

Step 17

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

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