Pre-Algebra Examples

$(3 x^{4} + 11 x^{3} - 20 x^{2} + 7 x + 35) \div (x + 5)$

Step 1

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

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

Step 2

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

Step 4

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

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

Step 5

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

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

Step 6

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

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

Step 7

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$4 x^{2}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$3 x^{3}$

$4 x^{2}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

$4 x^{3}$

$20 x^{2}$

Step 9

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

$3 x^{3}$

$4 x^{2}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

$4 x^{3}$

$20 x^{2}$

Step 10

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

$3 x^{3}$

$4 x^{2}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

$4 x^{3}$

$20 x^{2}$

$0$

Step 11

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

$3 x^{3}$

$4 x^{2}$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

$4 x^{3}$

$20 x^{2}$

$0$

$7 x$

$35$

Step 12

Divide the highest order term in the dividend $7 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

$4 x^{2}$

$0 x$

$7$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

$4 x^{3}$

$20 x^{2}$

$0$

$7 x$

$35$

Step 13

Multiply the new quotient term by the divisor.

$3 x^{3}$

$4 x^{2}$

$0 x$

$7$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

$4 x^{3}$

$20 x^{2}$

$0$

$7 x$

$35$

$7 x$

$35$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $7 x + 35$

$3 x^{3}$

$4 x^{2}$

$0 x$

$7$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

$4 x^{3}$

$20 x^{2}$

$0$

$7 x$

$35$

$7 x$

$35$

Step 15

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

$3 x^{3}$

$4 x^{2}$

$0 x$

$7$

$x$

$5$

$3 x^{4}$

$11 x^{3}$

$20 x^{2}$

$7 x$

$35$

$3 x^{4}$

$15 x^{3}$

$4 x^{3}$

$20 x^{2}$

$4 x^{3}$

$20 x^{2}$

$0$

$7 x$

$35$

$7 x$

$35$

$0$

Step 16

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

$3 x^{3} - 4 x^{2} + 0 x + 7$