Pre-Algebra Examples

$(2 x^{5} - 10^{3} + 2 x^{2} - 7 x + 2) \div (x^{2} + 1)$

Step 1

Expand $2 x^{5} - 10^{3} + 2 x^{2} - 7 x + 2$ .

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Step 1.1

Rewrite the exponentiation as a product.

$\frac{2 x^{5} - 1 \cdot 10 \cdot 10^{2} + 2 x^{2} - 7 x + 2}{x^{2} + 1}$

Step 1.2

Rewrite the exponentiation as a product.

$\frac{2 x^{5} - 1 \cdot (10 \cdot (10 \cdot 10)) + 2 x^{2} - 7 x + 2}{x^{2} + 1}$

Step 1.3

Remove parentheses.

$\frac{2 x^{5} - 1 \cdot (10 \cdot 10 \cdot 10) + 2 x^{2} - 7 x + 2}{x^{2} + 1}$

Step 1.4

Move parentheses.

$\frac{2 x^{5} - 1 \cdot (10 \cdot 10) \cdot 10 + 2 x^{2} - 7 x + 2}{x^{2} + 1}$

Step 1.5

Remove parentheses.

$\frac{2 x^{5} - 1 \cdot 10 \cdot 10 \cdot 10 + 2 x^{2} - 7 x + 2}{x^{2} + 1}$

Step 1.6

Multiply $- 1$ by $10$ .

$\frac{2 x^{5} - 10 \cdot 10 \cdot 10 + 2 x^{2} - 7 x + 2}{x^{2} + 1}$

Step 1.7

Multiply $- 10$ by $10$ .

$\frac{2 x^{5} - 100 \cdot 10 + 2 x^{2} - 7 x + 2}{x^{2} + 1}$

Step 1.8

Multiply $- 100$ by $10$ .

$\frac{2 x^{5} - 1000 + 2 x^{2} - 7 x + 2}{x^{2} + 1}$

Step 1.9

Move $- 1000$ .

$\frac{2 x^{5} + 2 x^{2} - 1000 - 7 x + 2}{x^{2} + 1}$

Step 1.10

Move $- 1000$ .

$\frac{2 x^{5} + 2 x^{2} - 7 x - 1000 + 2}{x^{2} + 1}$

Step 1.11

Add $- 1000$ and $2$ .

$\frac{2 x^{5} + 2 x^{2} - 7 x - 998}{x^{2} + 1}$

Step 2

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

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

Step 3

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

$2 x^{3}$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

Step 4

Multiply the new quotient term by the divisor.

$2 x^{3}$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

Step 5

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

$2 x^{3}$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

Step 6

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

$2 x^{3}$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

Step 7

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

$2 x^{3}$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$7 x$

Step 8

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

$2 x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$7 x$

Step 9

Multiply the new quotient term by the divisor.

$2 x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$7 x$

$2 x^{3}$

$0$

$2 x$

Step 10

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

$2 x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$7 x$

$2 x^{3}$

$0$

$2 x$

Step 11

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

$2 x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$7 x$

$2 x^{3}$

$0$

$2 x$

$2 x^{2}$

$5 x$

Step 12

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

$2 x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$7 x$

$2 x^{3}$

$0$

$2 x$

$2 x^{2}$

$5 x$

$998$

Step 13

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

$2 x^{3}$

$0 x^{2}$

$2 x$

$2$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$7 x$

$2 x^{3}$

$0$

$2 x$

$2 x^{2}$

$5 x$

$998$

Step 14

Multiply the new quotient term by the divisor.

$2 x^{3}$

$0 x^{2}$

$2 x$

$2$

$x^{2}$

$0 x$

$1$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$2 x^{2}$

$7 x$

$998$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$7 x$

$2 x^{3}$

$0$

$2 x$

$2 x^{2}$

$5 x$

$998$

$2 x^{2}$

$0$

$2$

Step 15

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

						$2 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$2$
$x^{2}$	+	$0 x$	+	$1$		$2 x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	+	$2 x^{2}$	-	$7 x$	-	$998$
					-	$2 x^{5}$	-	$0$	-	$2 x^{3}$
									-	$2 x^{3}$	+	$2 x^{2}$	-	$7 x$
									+	$2 x^{3}$	-	$0$	+	$2 x$
											+	$2 x^{2}$	-	$5 x$	-	$998$
											-	$2 x^{2}$	-	$0$	-	$2$

Step 16

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

						$2 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$2$
$x^{2}$	+	$0 x$	+	$1$		$2 x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	+	$2 x^{2}$	-	$7 x$	-	$998$
					-	$2 x^{5}$	-	$0$	-	$2 x^{3}$
									-	$2 x^{3}$	+	$2 x^{2}$	-	$7 x$
									+	$2 x^{3}$	-	$0$	+	$2 x$
											+	$2 x^{2}$	-	$5 x$	-	$998$
											-	$2 x^{2}$	-	$0$	-	$2$
													-	$5 x$	-	$1000$

Step 17

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

$2 x^{3} - 2 x + 2 + \frac{- 5 x - 1000}{x^{2} + 1}$