Algebra Examples

$\frac{2 x^{3} - 6 x^{2} + 7 x - 9}{2 x^{2} + 5}$

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 $0$ .

$2 x^{2}$

$0 x$

$5$

$2 x^{3}$

$6 x^{2}$

$7 x$

$9$

Step 1.2

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

$x$

$2 x^{2}$

$0 x$

$5$

$2 x^{3}$

$6 x^{2}$

$7 x$

$9$

Step 1.3

Multiply the new quotient term by the divisor.

$x$

$2 x^{2}$

$0 x$

$5$

$2 x^{3}$

$6 x^{2}$

$7 x$

$9$

$2 x^{3}$

$0$

$5 x$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 0 + 5 x$

						$x$
$2 x^{2}$	+	$0 x$	+	$5$		$2 x^{3}$	-	$6 x^{2}$	+	$7 x$	-	$9$
					-	$2 x^{3}$	-	$0$	-	$5 x$

Step 1.5

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

$x$

$2 x^{2}$

$0 x$

$5$

$2 x^{3}$

$6 x^{2}$

$7 x$

$9$

$2 x^{3}$

$0$

$5 x$

$6 x^{2}$

$2 x$

Step 1.6

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

						$x$
$2 x^{2}$	+	$0 x$	+	$5$		$2 x^{3}$	-	$6 x^{2}$	+	$7 x$	-	$9$
					-	$2 x^{3}$	-	$0$	-	$5 x$
							-	$6 x^{2}$	+	$2 x$	-	$9$

Step 1.7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $2 x^{2}$ .

						$x$	-	$3$
$2 x^{2}$	+	$0 x$	+	$5$		$2 x^{3}$	-	$6 x^{2}$	+	$7 x$	-	$9$
					-	$2 x^{3}$	-	$0$	-	$5 x$
							-	$6 x^{2}$	+	$2 x$	-	$9$

Step 1.8

Multiply the new quotient term by the divisor.

						$x$	-	$3$
$2 x^{2}$	+	$0 x$	+	$5$		$2 x^{3}$	-	$6 x^{2}$	+	$7 x$	-	$9$
					-	$2 x^{3}$	-	$0$	-	$5 x$
							-	$6 x^{2}$	+	$2 x$	-	$9$
							-	$6 x^{2}$	+	$0$	-	$15$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 0 - 15$

						$x$	-	$3$
$2 x^{2}$	+	$0 x$	+	$5$		$2 x^{3}$	-	$6 x^{2}$	+	$7 x$	-	$9$
					-	$2 x^{3}$	-	$0$	-	$5 x$
							-	$6 x^{2}$	+	$2 x$	-	$9$
							+	$6 x^{2}$	-	$0$	+	$15$

Step 1.10

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

						$x$	-	$3$
$2 x^{2}$	+	$0 x$	+	$5$		$2 x^{3}$	-	$6 x^{2}$	+	$7 x$	-	$9$
					-	$2 x^{3}$	-	$0$	-	$5 x$
							-	$6 x^{2}$	+	$2 x$	-	$9$
							+	$6 x^{2}$	-	$0$	+	$15$
									+	$2 x$	+	$6$

Step 1.11

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

$x - 3 + \frac{2 x + 6}{2 x^{2} + 5}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$2 x + 6$