Pre-Algebra Examples

$(x^{4} - 3 x^{3} - 28 x^{2} + 59 x + 6) \div (x - 6)$

Step 1

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

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

Step 2

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

$x^{3}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

Step 3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 6 x^{3}$

$x^{3}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

Step 5

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

$x^{3}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

Step 6

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

$x^{3}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

Step 7

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

$x^{3}$

$3 x^{2}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

Step 9

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

$x^{3}$

$3 x^{2}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

Step 10

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

$x^{3}$

$3 x^{2}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

Step 11

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

$x^{3}$

$3 x^{2}$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

Step 12

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

$x^{3}$

$3 x^{2}$

$10 x$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

Step 13

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$10 x$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$60 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} + 60 x$

$x^{3}$

$3 x^{2}$

$10 x$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$60 x$

Step 15

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

$x^{3}$

$3 x^{2}$

$10 x$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$60 x$

$x$

Step 16

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

$x^{3}$

$3 x^{2}$

$10 x$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$60 x$

$x$

$6$

Step 17

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

$x^{3}$

$3 x^{2}$

$10 x$

$1$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$60 x$

$x$

$6$

Step 18

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$10 x$

$1$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$60 x$

$x$

$6$

$x$

$6$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- x + 6$

$x^{3}$

$3 x^{2}$

$10 x$

$1$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$60 x$

$x$

$6$

$x$

$6$

Step 20

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

$x^{3}$

$3 x^{2}$

$10 x$

$1$

$x$

$6$

$x^{4}$

$3 x^{3}$

$28 x^{2}$

$59 x$

$6$

$x^{4}$

$6 x^{3}$

$3 x^{3}$

$28 x^{2}$

$3 x^{3}$

$18 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$60 x$

$x$

$6$

$x$

$6$

$0$

Step 21

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

$x^{3} + 3 x^{2} - 10 x - 1$