Algebra Examples

\frac{7 x^{2} - 4 x - 3}{x + 2}

$\frac{7 x^{2} - 4 x - 3}{x + 2}$

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

$0$ .

x

$x$

2

$2$

7 x^{2}

$7 x^{2}$

4 x

$4 x$

3

$3$

Step 1.2

Divide the highest order term in the dividend

7 x^{2}

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

x

$x$ .

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

Step 1.3

Multiply the new quotient term by the divisor.

				$7 x$ $7 x$
$x$ $x$	+	$2$ $2$		$7 x^{2}$ $7 x^{2}$	-	$4 x$ $4 x$	-	$3$ $3$
			+	$7 x^{2}$ $7 x^{2}$	+	$14 x$ $14 x$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in

7 x^{2} + 14 x

$7 x^{2} + 14 x$

				$7 x$ $7 x$
$x$ $x$	+	$2$ $2$		$7 x^{2}$ $7 x^{2}$	-	$4 x$ $4 x$	-	$3$ $3$
			-	$7 x^{2}$ $7 x^{2}$	-	$14 x$ $14 x$

Step 1.5

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

				$7 x$ $7 x$
$x$ $x$	+	$2$ $2$		$7 x^{2}$ $7 x^{2}$	-	$4 x$ $4 x$	-	$3$ $3$
			-	$7 x^{2}$ $7 x^{2}$	-	$14 x$ $14 x$
					-	$18 x$ $18 x$

Step 1.6

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

				$7 x$ $7 x$
$x$ $x$	+	$2$ $2$		$7 x^{2}$ $7 x^{2}$	-	$4 x$ $4 x$	-	$3$ $3$
			-	$7 x^{2}$ $7 x^{2}$	-	$14 x$ $14 x$
					-	$18 x$ $18 x$	-	$3$ $3$

Step 1.7

Divide the highest order term in the dividend

- 18 x

$- 18 x$ by the highest order term in divisor

x

$x$ .

				$7 x$ $7 x$	-	$18$ $18$
$x$ $x$	+	$2$ $2$		$7 x^{2}$ $7 x^{2}$	-	$4 x$ $4 x$	-	$3$ $3$
			-	$7 x^{2}$ $7 x^{2}$	-	$14 x$ $14 x$
					-	$18 x$ $18 x$	-	$3$ $3$

Step 1.8

Multiply the new quotient term by the divisor.

				$7 x$ $7 x$	-	$18$ $18$
$x$ $x$	+	$2$ $2$		$7 x^{2}$ $7 x^{2}$	-	$4 x$ $4 x$	-	$3$ $3$
			-	$7 x^{2}$ $7 x^{2}$	-	$14 x$ $14 x$
					-	$18 x$ $18 x$	-	$3$ $3$
					-	$18 x$ $18 x$	-	$36$ $36$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in

- 18 x - 36

$- 18 x - 36$

				$7 x$ $7 x$	-	$18$ $18$
$x$ $x$	+	$2$ $2$		$7 x^{2}$ $7 x^{2}$	-	$4 x$ $4 x$	-	$3$ $3$
			-	$7 x^{2}$ $7 x^{2}$	-	$14 x$ $14 x$
					-	$18 x$ $18 x$	-	$3$ $3$
					+	$18 x$ $18 x$	+	$36$ $36$

Step 1.10

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

				$7 x$ $7 x$	-	$18$ $18$
$x$ $x$	+	$2$ $2$		$7 x^{2}$ $7 x^{2}$	-	$4 x$ $4 x$	-	$3$ $3$
			-	$7 x^{2}$ $7 x^{2}$	-	$14 x$ $14 x$
					-	$18 x$ $18 x$	-	$3$ $3$
					+	$18 x$ $18 x$	+	$36$ $36$
							+	$33$ $33$

Step 1.11

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

7 x - 18 + \frac{33}{x + 2}

$7 x - 18 + \frac{33}{x + 2}$

7 x - 18 + \frac{33}{x + 2}

$7 x - 18 + \frac{33}{x + 2}$

Step 2

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

33

$33$

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