Algebra Examples

$(5 x^{4} + 2 x^{3} - 9 x + 12) \div (x^{2} - 3 x + 4)$

Step 1

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

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

Step 2

Divide the highest order term in the dividend $5 x^{4}$ by the highest order term in divisor $x^{2}$ .

$5 x^{2}$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

Step 3

Multiply the new quotient term by the divisor.

$5 x^{2}$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} - 15 x^{3} + 20 x^{2}$

$5 x^{2}$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

Step 5

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

$5 x^{2}$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

Step 6

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

$5 x^{2}$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

Step 7

Divide the highest order term in the dividend $17 x^{3}$ by the highest order term in divisor $x^{2}$ .

$5 x^{2}$

$17 x$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

Step 8

Multiply the new quotient term by the divisor.

$5 x^{2}$

$17 x$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

$17 x^{3}$

$51 x^{2}$

$68 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $17 x^{3} - 51 x^{2} + 68 x$

$5 x^{2}$

$17 x$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

$17 x^{3}$

$51 x^{2}$

$68 x$

Step 10

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

$5 x^{2}$

$17 x$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

$17 x^{3}$

$51 x^{2}$

$68 x$

$31 x^{2}$

$77 x$

Step 11

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

$5 x^{2}$

$17 x$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

$17 x^{3}$

$51 x^{2}$

$68 x$

$31 x^{2}$

$77 x$

$12$

Step 12

Divide the highest order term in the dividend $31 x^{2}$ by the highest order term in divisor $x^{2}$ .

$5 x^{2}$

$17 x$

$31$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

$17 x^{3}$

$51 x^{2}$

$68 x$

$31 x^{2}$

$77 x$

$12$

Step 13

Multiply the new quotient term by the divisor.

$5 x^{2}$

$17 x$

$31$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

$17 x^{3}$

$51 x^{2}$

$68 x$

$31 x^{2}$

$77 x$

$12$

$31 x^{2}$

$93 x$

$124$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $31 x^{2} - 93 x + 124$

$5 x^{2}$

$17 x$

$31$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

$17 x^{3}$

$51 x^{2}$

$68 x$

$31 x^{2}$

$77 x$

$12$

$31 x^{2}$

$93 x$

$124$

Step 15

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

$5 x^{2}$

$17 x$

$31$

$x^{2}$

$3 x$

$4$

$5 x^{4}$

$2 x^{3}$

$0 x^{2}$

$9 x$

$12$

$5 x^{4}$

$15 x^{3}$

$20 x^{2}$

$17 x^{3}$

$20 x^{2}$

$9 x$

$17 x^{3}$

$51 x^{2}$

$68 x$

$31 x^{2}$

$77 x$

$12$

$31 x^{2}$

$93 x$

$124$

$16 x$

$112$

Step 16

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

$5 x^{2} + 17 x + 31 + \frac{16 x - 112}{x^{2} - 3 x + 4}$