Algebra Examples

$\frac{- 4 t^{4} + 18 t^{3} + 8 t^{2} - 60 t - 40}{2 t^{2} - 4 t - 4}$

Step 1

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

$2 t^{2}$

$4 t$

$4$

$4 t^{4}$

$18 t^{3}$

$8 t^{2}$

$60 t$

$40$

Step 2

Divide the highest order term in the dividend $- 4 t^{4}$ by the highest order term in divisor $2 t^{2}$ .

						-	$2 t^{2}$
	$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$

Step 3

Multiply the new quotient term by the divisor.

$2 t^{2}$

$4 t$

$4$

$4 t^{4}$

$18 t^{3}$

$8 t^{2}$

$60 t$

$40$

$4 t^{4}$

$8 t^{3}$

$8 t^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 t^{4} + 8 t^{3} + 8 t^{2}$

					-	$2 t^{2}$
$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$
					+	$4 t^{4}$	-	$8 t^{3}$	-	$8 t^{2}$

Step 5

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

					-	$2 t^{2}$
$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$
					+	$4 t^{4}$	-	$8 t^{3}$	-	$8 t^{2}$
							+	$10 t^{3}$	+	$0$

Step 6

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

					-	$2 t^{2}$
$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$
					+	$4 t^{4}$	-	$8 t^{3}$	-	$8 t^{2}$
							+	$10 t^{3}$	+	$0$	-	$60 t$	-	$40$

Step 7

Divide the highest order term in the dividend $10 t^{3}$ by the highest order term in divisor $2 t^{2}$ .

					-	$2 t^{2}$	+	$5 t$
$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$
					+	$4 t^{4}$	-	$8 t^{3}$	-	$8 t^{2}$
							+	$10 t^{3}$	+	$0$	-	$60 t$	-	$40$

Step 8

Multiply the new quotient term by the divisor.

					-	$2 t^{2}$	+	$5 t$
$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$
					+	$4 t^{4}$	-	$8 t^{3}$	-	$8 t^{2}$
							+	$10 t^{3}$	+	$0$	-	$60 t$	-	$40$
							+	$10 t^{3}$	-	$20 t^{2}$	-	$20 t$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $10 t^{3} - 20 t^{2} - 20 t$

$2 t^{2}$

$5 t$

$2 t^{2}$

$4 t$

$4$

$4 t^{4}$

$18 t^{3}$

$8 t^{2}$

$60 t$

$40$

$4 t^{4}$

$8 t^{3}$

$8 t^{2}$

$10 t^{3}$

$0$

$60 t$

$40$

$10 t^{3}$

$20 t^{2}$

$20 t$

Step 10

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

$2 t^{2}$

$5 t$

$2 t^{2}$

$4 t$

$4$

$4 t^{4}$

$18 t^{3}$

$8 t^{2}$

$60 t$

$40$

$4 t^{4}$

$8 t^{3}$

$8 t^{2}$

$10 t^{3}$

$0$

$60 t$

$40$

$10 t^{3}$

$20 t^{2}$

$20 t$

$20 t^{2}$

$40 t$

Step 11

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

					-	$2 t^{2}$	+	$5 t$
$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$
					+	$4 t^{4}$	-	$8 t^{3}$	-	$8 t^{2}$
							+	$10 t^{3}$	+	$0$	-	$60 t$	-	$40$
							-	$10 t^{3}$	+	$20 t^{2}$	+	$20 t$
									+	$20 t^{2}$	-	$40 t$	-	$40$

Step 12

Divide the highest order term in the dividend $20 t^{2}$ by the highest order term in divisor $2 t^{2}$ .

					-	$2 t^{2}$	+	$5 t$	+	$10$
$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$
					+	$4 t^{4}$	-	$8 t^{3}$	-	$8 t^{2}$
							+	$10 t^{3}$	+	$0$	-	$60 t$	-	$40$
							-	$10 t^{3}$	+	$20 t^{2}$	+	$20 t$
									+	$20 t^{2}$	-	$40 t$	-	$40$

Step 13

Multiply the new quotient term by the divisor.

					-	$2 t^{2}$	+	$5 t$	+	$10$
$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$
					+	$4 t^{4}$	-	$8 t^{3}$	-	$8 t^{2}$
							+	$10 t^{3}$	+	$0$	-	$60 t$	-	$40$
							-	$10 t^{3}$	+	$20 t^{2}$	+	$20 t$
									+	$20 t^{2}$	-	$40 t$	-	$40$
									+	$20 t^{2}$	-	$40 t$	-	$40$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $20 t^{2} - 40 t - 40$

$2 t^{2}$

$5 t$

$10$

$2 t^{2}$

$4 t$

$4$

$4 t^{4}$

$18 t^{3}$

$8 t^{2}$

$60 t$

$40$

$4 t^{4}$

$8 t^{3}$

$8 t^{2}$

$10 t^{3}$

$0$

$60 t$

$40$

$10 t^{3}$

$20 t^{2}$

$20 t$

$20 t^{2}$

$40 t$

$40$

$20 t^{2}$

$40 t$

$40$

Step 15

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

					-	$2 t^{2}$	+	$5 t$	+	$10$
$2 t^{2}$	-	$4 t$	-	$4$	-	$4 t^{4}$	+	$18 t^{3}$	+	$8 t^{2}$	-	$60 t$	-	$40$
					+	$4 t^{4}$	-	$8 t^{3}$	-	$8 t^{2}$
							+	$10 t^{3}$	+	$0$	-	$60 t$	-	$40$
							-	$10 t^{3}$	+	$20 t^{2}$	+	$20 t$
									+	$20 t^{2}$	-	$40 t$	-	$40$
									-	$20 t^{2}$	+	$40 t$	+	$40$
														$0$

Step 16

Since the remander is $0$ , the final answer is the quotient.

$- 2 t^{2} + 5 t + 10$