Algebra Examples

$\frac{- x^{3} + 4 x^{2} + 5 x - 4}{x + 3}$

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 $0$ .

$x$

$3$

$x^{3}$

$4 x^{2}$

$5 x$

$4$

Step 1.2

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

				-	$x^{2}$
	$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$

Step 1.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			-	$x^{3}$	-	$3 x^{2}$

Step 1.4

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

			-	$x^{2}$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$

Step 1.5

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

			-	$x^{2}$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$

Step 1.6

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

			-	$x^{2}$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$

Step 1.7

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$7 x$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$

Step 1.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$7 x$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$
					+	$7 x^{2}$	+	$21 x$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 21 x$

			-	$x^{2}$	+	$7 x$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$
					-	$7 x^{2}$	-	$21 x$

Step 1.10

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

			-	$x^{2}$	+	$7 x$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$
					-	$7 x^{2}$	-	$21 x$
							-	$16 x$

Step 1.11

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

			-	$x^{2}$	+	$7 x$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$
					-	$7 x^{2}$	-	$21 x$
							-	$16 x$	-	$4$

Step 1.12

Divide the highest order term in the dividend $- 16 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$7 x$	-	$16$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$
					-	$7 x^{2}$	-	$21 x$
							-	$16 x$	-	$4$

Step 1.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$7 x$	-	$16$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$
					-	$7 x^{2}$	-	$21 x$
							-	$16 x$	-	$4$
							-	$16 x$	-	$48$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x - 48$

			-	$x^{2}$	+	$7 x$	-	$16$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$
					-	$7 x^{2}$	-	$21 x$
							-	$16 x$	-	$4$
							+	$16 x$	+	$48$

Step 1.15

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

			-	$x^{2}$	+	$7 x$	-	$16$
$x$	+	$3$	-	$x^{3}$	+	$4 x^{2}$	+	$5 x$	-	$4$
			+	$x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	+	$5 x$
					-	$7 x^{2}$	-	$21 x$
							-	$16 x$	-	$4$
							+	$16 x$	+	$48$
									+	$44$

Step 1.16

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

$- x^{2} + 7 x - 16 + \frac{44}{x + 3}$

Step 2

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

$44$