Pre-Algebra Examples

$(t^{3} - 3 t^{2} + 7 t + 12) \div (t + 3)$

Step 1

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

$t$

$3$

$t^{3}$

$3 t^{2}$

$7 t$

$12$

Step 2

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

					$t^{2}$
	$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$

Step 3

Multiply the new quotient term by the divisor.

				$t^{2}$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			+	$t^{3}$	+	$3 t^{2}$

Step 4

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

				$t^{2}$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$

Step 5

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

				$t^{2}$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$

Step 6

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

				$t^{2}$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$

Step 7

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

				$t^{2}$	-	$6 t$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$

Step 8

Multiply the new quotient term by the divisor.

				$t^{2}$	-	$6 t$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$
					-	$6 t^{2}$	-	$18 t$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 t^{2} - 18 t$

				$t^{2}$	-	$6 t$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$
					+	$6 t^{2}$	+	$18 t$

Step 10

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

				$t^{2}$	-	$6 t$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$
					+	$6 t^{2}$	+	$18 t$
							+	$25 t$

Step 11

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

				$t^{2}$	-	$6 t$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$
					+	$6 t^{2}$	+	$18 t$
							+	$25 t$	+	$12$

Step 12

Divide the highest order term in the dividend $25 t$ by the highest order term in divisor $t$ .

				$t^{2}$	-	$6 t$	+	$25$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$
					+	$6 t^{2}$	+	$18 t$
							+	$25 t$	+	$12$

Step 13

Multiply the new quotient term by the divisor.

				$t^{2}$	-	$6 t$	+	$25$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$
					+	$6 t^{2}$	+	$18 t$
							+	$25 t$	+	$12$
							+	$25 t$	+	$75$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $25 t + 75$

				$t^{2}$	-	$6 t$	+	$25$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$
					+	$6 t^{2}$	+	$18 t$
							+	$25 t$	+	$12$
							-	$25 t$	-	$75$

Step 15

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

				$t^{2}$	-	$6 t$	+	$25$
$t$	+	$3$		$t^{3}$	-	$3 t^{2}$	+	$7 t$	+	$12$
			-	$t^{3}$	-	$3 t^{2}$
					-	$6 t^{2}$	+	$7 t$
					+	$6 t^{2}$	+	$18 t$
							+	$25 t$	+	$12$
							-	$25 t$	-	$75$
									-	$63$

Step 16

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

$t^{2} - 6 t + 25 - \frac{63}{t + 3}$