Algebra Examples

$(5 x^{3} - 14 - 20 x + x^{4}) \div (x^{2} - x - 3)$

Step 1

Expand $5 x^{3} - 14 - 20 x + x^{4}$ .

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Step 1.1

Move $- 14$ .

$\frac{5 x^{3} - 20 x - 14 + x^{4}}{x^{2} - x - 3}$

Step 1.2

Move $- 14$ .

$\frac{5 x^{3} - 20 x + x^{4} - 14}{x^{2} - x - 3}$

Step 1.3

Move $- 20 x$ .

$\frac{5 x^{3} + x^{4} - 20 x - 14}{x^{2} - x - 3}$

Step 1.4

Reorder $5 x^{3}$ and $x^{4}$ .

$\frac{x^{4} + 5 x^{3} - 20 x - 14}{x^{2} - x - 3}$

Step 2

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

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

Step 3

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

							$x^{2}$
	$x^{2}$	-	$x$	-	$3$		$x^{4}$	+	$5 x^{3}$	+	$0 x^{2}$	-	$20 x$	-	$14$

Step 4

Multiply the new quotient term by the divisor.

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

Step 5

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

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

Step 6

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

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

Step 7

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

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

Step 8

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$6 x$

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

Step 9

Multiply the new quotient term by the divisor.

$x^{2}$

$6 x$

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

$6 x^{3}$

$6 x^{2}$

$18 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} - 6 x^{2} - 18 x$

$x^{2}$

$6 x$

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

$6 x^{3}$

$6 x^{2}$

$18 x$

Step 11

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

$x^{2}$

$6 x$

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

$6 x^{3}$

$6 x^{2}$

$18 x$

$9 x^{2}$

$2 x$

Step 12

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

$x^{2}$

$6 x$

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

$6 x^{3}$

$6 x^{2}$

$18 x$

$9 x^{2}$

$2 x$

$14$

Step 13

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$6 x$

$9$

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

$6 x^{3}$

$6 x^{2}$

$18 x$

$9 x^{2}$

$2 x$

$14$

Step 14

Multiply the new quotient term by the divisor.

$x^{2}$

$6 x$

$9$

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

$6 x^{3}$

$6 x^{2}$

$18 x$

$9 x^{2}$

$2 x$

$14$

$9 x^{2}$

$9 x$

$27$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} - 9 x - 27$

$x^{2}$

$6 x$

$9$

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

$6 x^{3}$

$6 x^{2}$

$18 x$

$9 x^{2}$

$2 x$

$14$

$9 x^{2}$

$9 x$

$27$

Step 16

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

$x^{2}$

$6 x$

$9$

$x^{2}$

$x$

$3$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$14$

$x^{4}$

$x^{3}$

$3 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x$

$6 x^{3}$

$6 x^{2}$

$18 x$

$9 x^{2}$

$2 x$

$14$

$9 x^{2}$

$9 x$

$27$

$7 x$

$13$

Step 17

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

$x^{2} + 6 x + 9 + \frac{7 x + 13}{x^{2} - x - 3}$