Algebra Examples

$\frac{3 x - 4 x^{3} + 6 x^{4} + 1}{x + 3}$

Step 1

Expand $3 x - 4 x^{3} + 6 x^{4} + 1$ .

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Step 1.1

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

$\frac{- 4 x^{3} + 3 x + 6 x^{4} + 1}{x + 3}$

Step 1.2

Move $3 x$ .

$\frac{- 4 x^{3} + 6 x^{4} + 3 x + 1}{x + 3}$

Step 1.3

Reorder $- 4 x^{3}$ and $6 x^{4}$ .

$\frac{6 x^{4} - 4 x^{3} + 3 x + 1}{x + 3}$

Step 2

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

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

Step 3

Divide the highest order term in the dividend $6 x^{4}$ by the highest order term in divisor $x$ .

$6 x^{3}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

Step 4

Multiply the new quotient term by the divisor.

$6 x^{3}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{4} + 18 x^{3}$

$6 x^{3}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

Step 6

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

$6 x^{3}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

Step 7

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

$6 x^{3}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

Step 8

Divide the highest order term in the dividend $- 22 x^{3}$ by the highest order term in divisor $x$ .

$6 x^{3}$

$22 x^{2}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

Step 9

Multiply the new quotient term by the divisor.

$6 x^{3}$

$22 x^{2}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 22 x^{3} - 66 x^{2}$

$6 x^{3}$

$22 x^{2}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

Step 11

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

$6 x^{3}$

$22 x^{2}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

Step 12

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

$6 x^{3}$

$22 x^{2}$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

Step 13

Divide the highest order term in the dividend $66 x^{2}$ by the highest order term in divisor $x$ .

$6 x^{3}$

$22 x^{2}$

$66 x$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

Step 14

Multiply the new quotient term by the divisor.

$6 x^{3}$

$22 x^{2}$

$66 x$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

$66 x^{2}$

$198 x$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $66 x^{2} + 198 x$

$6 x^{3}$

$22 x^{2}$

$66 x$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

$66 x^{2}$

$198 x$

Step 16

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

$6 x^{3}$

$22 x^{2}$

$66 x$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

$66 x^{2}$

$198 x$

$195 x$

Step 17

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

$6 x^{3}$

$22 x^{2}$

$66 x$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

$66 x^{2}$

$198 x$

$195 x$

$1$

Step 18

Divide the highest order term in the dividend $- 195 x$ by the highest order term in divisor $x$ .

$6 x^{3}$

$22 x^{2}$

$66 x$

$195$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

$66 x^{2}$

$198 x$

$195 x$

$1$

Step 19

Multiply the new quotient term by the divisor.

$6 x^{3}$

$22 x^{2}$

$66 x$

$195$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

$66 x^{2}$

$198 x$

$195 x$

$1$

$195 x$

$585$

Step 20

The expression needs to be subtracted from the dividend, so change all the signs in $- 195 x - 585$

$6 x^{3}$

$22 x^{2}$

$66 x$

$195$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

$66 x^{2}$

$198 x$

$195 x$

$1$

$195 x$

$585$

Step 21

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

$6 x^{3}$

$22 x^{2}$

$66 x$

$195$

$x$

$3$

$6 x^{4}$

$4 x^{3}$

$0 x^{2}$

$3 x$

$1$

$6 x^{4}$

$18 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$66 x^{2}$

$3 x$

$66 x^{2}$

$198 x$

$195 x$

$1$

$195 x$

$585$

$586$

Step 22

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

$6 x^{3} - 22 x^{2} + 66 x - 195 + \frac{586}{x + 3}$