Finite Math Examples

$10 x^{4} + 27 x^{3} + 13 x^{2} + 16 x - 15$ , $2 x + 5$

Step 1

Divide the first expression by the second expression.

$\frac{10 x^{4} + 27 x^{3} + 13 x^{2} + 16 x - 15}{2 x + 5}$

Step 2

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

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

Step 3

Divide the highest order term in the dividend $10 x^{4}$ by the highest order term in divisor $2 x$ .

$5 x^{3}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

Step 4

Multiply the new quotient term by the divisor.

$5 x^{3}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{4} + 25 x^{3}$

$5 x^{3}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

Step 6

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

$5 x^{3}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

Step 7

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

$5 x^{3}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

Step 8

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

$5 x^{3}$

$x^{2}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

Step 9

Multiply the new quotient term by the divisor.

$5 x^{3}$

$x^{2}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

Step 10

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

$5 x^{3}$

$x^{2}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

Step 11

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

$5 x^{3}$

$x^{2}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

Step 12

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

$5 x^{3}$

$x^{2}$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

Step 13

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $2 x$ .

$5 x^{3}$

$x^{2}$

$4 x$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

Step 14

Multiply the new quotient term by the divisor.

$5 x^{3}$

$x^{2}$

$4 x$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

$8 x^{2}$

$20 x$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} + 20 x$

$5 x^{3}$

$x^{2}$

$4 x$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

$8 x^{2}$

$20 x$

Step 16

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

$5 x^{3}$

$x^{2}$

$4 x$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

$8 x^{2}$

$20 x$

$4 x$

Step 17

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

$5 x^{3}$

$x^{2}$

$4 x$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

$8 x^{2}$

$20 x$

$4 x$

$15$

Step 18

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $2 x$ .

$5 x^{3}$

$x^{2}$

$4 x$

$2$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

$8 x^{2}$

$20 x$

$4 x$

$15$

Step 19

Multiply the new quotient term by the divisor.

$5 x^{3}$

$x^{2}$

$4 x$

$2$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

$8 x^{2}$

$20 x$

$4 x$

$15$

$4 x$

$10$

Step 20

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 10$

$5 x^{3}$

$x^{2}$

$4 x$

$2$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

$8 x^{2}$

$20 x$

$4 x$

$15$

$4 x$

$10$

Step 21

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

$5 x^{3}$

$x^{2}$

$4 x$

$2$

$2 x$

$5$

$10 x^{4}$

$27 x^{3}$

$13 x^{2}$

$16 x$

$15$

$10 x^{4}$

$25 x^{3}$

$2 x^{3}$

$13 x^{2}$

$2 x^{3}$

$5 x^{2}$

$8 x^{2}$

$16 x$

$8 x^{2}$

$20 x$

$4 x$

$15$

$4 x$

$10$

$5$

Step 22

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

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