Divide Using Long Polynomial Division (10x^4-20x^3+25x^2)÷(5x^2)

$(10 x^{4} - 20 x^{3} + 25 x^{2}) \div (5 x^{2})$

Step 1

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

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $10 x^{4}$ by the highest order term in divisor $5 x^{2}$ .

$2 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{4} + 0 + 0$

$2 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

Step 5

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

$2 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

$20 x^{3}$

$25 x^{2}$

Step 6

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

$2 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

$20 x^{3}$

$25 x^{2}$

$0 x$

Step 7

Divide the highest order term in the dividend $- 20 x^{3}$ by the highest order term in divisor $5 x^{2}$ .

$2 x^{2}$

$4 x$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

$20 x^{3}$

$25 x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{2}$

$4 x$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

$20 x^{3}$

$25 x^{2}$

$0 x$

$20 x^{3}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x^{3} + 0 + 0$

						$2 x^{2}$	-	$4 x$
$5 x^{2}$	+	$0 x$	+	$0$		$10 x^{4}$	-	$20 x^{3}$	+	$25 x^{2}$	+	$0 x$	+	$0$
					-	$10 x^{4}$	-	$0$	-	$0$
							-	$20 x^{3}$	+	$25 x^{2}$	+	$0 x$
							+	$20 x^{3}$	-	$0$	-	$0$

Step 10

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

$2 x^{2}$

$4 x$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

$20 x^{3}$

$25 x^{2}$

$0 x$

$20 x^{3}$

$0$

$25 x^{2}$

$0$

Step 11

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

$2 x^{2}$

$4 x$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

$20 x^{3}$

$25 x^{2}$

$0 x$

$20 x^{3}$

$0$

$25 x^{2}$

$0$

Step 12

Divide the highest order term in the dividend $25 x^{2}$ by the highest order term in divisor $5 x^{2}$ .

$2 x^{2}$

$4 x$

$5$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

$20 x^{3}$

$25 x^{2}$

$0 x$

$20 x^{3}$

$0$

$25 x^{2}$

$0$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{2}$

$4 x$

$5$

$5 x^{2}$

$0 x$

$0$

$10 x^{4}$

$20 x^{3}$

$25 x^{2}$

$0 x$

$0$

$10 x^{4}$

$0$

$20 x^{3}$

$25 x^{2}$

$0 x$

$20 x^{3}$

$0$

$25 x^{2}$

$0$

$25 x^{2}$

$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $25 x^{2} + 0 + 0$

						$2 x^{2}$	-	$4 x$	+	$5$
$5 x^{2}$	+	$0 x$	+	$0$		$10 x^{4}$	-	$20 x^{3}$	+	$25 x^{2}$	+	$0 x$	+	$0$
					-	$10 x^{4}$	-	$0$	-	$0$
							-	$20 x^{3}$	+	$25 x^{2}$	+	$0 x$
							+	$20 x^{3}$	-	$0$	-	$0$
									+	$25 x^{2}$	+	$0$	+	$0$
									-	$25 x^{2}$	-	$0$	-	$0$

Step 15

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

						$2 x^{2}$	-	$4 x$	+	$5$
$5 x^{2}$	+	$0 x$	+	$0$		$10 x^{4}$	-	$20 x^{3}$	+	$25 x^{2}$	+	$0 x$	+	$0$
					-	$10 x^{4}$	-	$0$	-	$0$
							-	$20 x^{3}$	+	$25 x^{2}$	+	$0 x$
							+	$20 x^{3}$	-	$0$	-	$0$
									+	$25 x^{2}$	+	$0$	+	$0$
									-	$25 x^{2}$	-	$0$	-	$0$
														$0$

Step 16

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

$2 x^{2} - 4 x + 5$

Divide Using Long Polynomial Division (10x^4-20x^3+25x^2)÷(5x^2)

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