Algebra Examples

$(2 x^{4} - 3 x^{3} - 3 x^{2} + 7 x - 3) \div (x^{2} - 2 x + 1)$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

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

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

Step 1.2

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

							$2 x^{2}$
	$x^{2}$	-	$2 x$	+	$1$		$2 x^{4}$	-	$3 x^{3}$	-	$3 x^{2}$	+	$7 x$	-	$3$

Step 1.3

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

Step 1.4

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

$2 x^{2}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

Step 1.5

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

$2 x^{2}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

Step 1.6

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

$2 x^{2}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

Step 1.7

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

$2 x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

Step 1.8

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

$x^{3}$

$2 x^{2}$

$x$

Step 1.9

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

$2 x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

$x^{3}$

$2 x^{2}$

$x$

Step 1.10

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

$2 x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

$x^{3}$

$2 x^{2}$

$x$

$3 x^{2}$

$6 x$

Step 1.11

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

$2 x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

$x^{3}$

$2 x^{2}$

$x$

$3 x^{2}$

$6 x$

$3$

Step 1.12

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

$2 x^{2}$

$x$

$3$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

$x^{3}$

$2 x^{2}$

$x$

$3 x^{2}$

$6 x$

$3$

Step 1.13

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x$

$3$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

$x^{3}$

$2 x^{2}$

$x$

$3 x^{2}$

$6 x$

$3$

$3 x^{2}$

$6 x$

$3$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{2} + 6 x - 3$

$2 x^{2}$

$x$

$3$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

$x^{3}$

$2 x^{2}$

$x$

$3 x^{2}$

$6 x$

$3$

$3 x^{2}$

$6 x$

$3$

Step 1.15

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

$2 x^{2}$

$x$

$3$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$3 x^{2}$

$7 x$

$3$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$x^{3}$

$5 x^{2}$

$7 x$

$x^{3}$

$2 x^{2}$

$x$

$3 x^{2}$

$6 x$

$3$

$3 x^{2}$

$6 x$

$3$

$0$

Step 1.16

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

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

Step 2

Since the final term in the resulting expression is not a fraction, the remainder is $0$ .

$0$