Algebra Examples

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

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

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

$0$ .

3 x^{3}

$3 x^{3}$

2 x^{2} y

$2 x^{2} y$

y^{2} x

$y^{2} x$

0

$0$

9 x^{4}

$9 x^{4}$

3 x^{3} y

$3 x^{3} y$

5 x^{2} y^{2}

$5 x^{2} y^{2}$

y^{3} x

$y^{3} x$

0

$0$

Step 2

Divide the highest order term in the dividend

9 x^{4}

$9 x^{4}$ by the highest order term in divisor

3 x^{3}

$3 x^{3}$ .

3 x

$3 x$

3 x^{3}

$3 x^{3}$

2 x^{2} y

$2 x^{2} y$

y^{2} x

$y^{2} x$

0

$0$

9 x^{4}

$9 x^{4}$

3 x^{3} y

$3 x^{3} y$

5 x^{2} y^{2}

$5 x^{2} y^{2}$

y^{3} x

$y^{3} x$

0

$0$

Step 3

Multiply the new quotient term by the divisor.

3 x

$3 x$

3 x^{3}

$3 x^{3}$

2 x^{2} y

$2 x^{2} y$

y^{2} x

$y^{2} x$

0

$0$

9 x^{4}

$9 x^{4}$

3 x^{3} y

$3 x^{3} y$

5 x^{2} y^{2}

$5 x^{2} y^{2}$

y^{3} x

$y^{3} x$

0

$0$

9 x^{4}

$9 x^{4}$

6 x^{3} y

$6 x^{3} y$

3 x^{2} y^{2}

$3 x^{2} y^{2}$

0

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in

9 x^{4} + 6 x^{3} y - 3 x^{2} y^{2} + 0

$9 x^{4} + 6 x^{3} y - 3 x^{2} y^{2} + 0$

3 x

$3 x$

3 x^{3}

$3 x^{3}$

2 x^{2} y

$2 x^{2} y$

y^{2} x

$y^{2} x$

0

$0$

9 x^{4}

$9 x^{4}$

3 x^{3} y

$3 x^{3} y$

5 x^{2} y^{2}

$5 x^{2} y^{2}$

y^{3} x

$y^{3} x$

0

$0$

9 x^{4}

$9 x^{4}$

6 x^{3} y

$6 x^{3} y$

3 x^{2} y^{2}

$3 x^{2} y^{2}$

0

$0$

Step 5

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

3 x

$3 x$

3 x^{3}

$3 x^{3}$

2 x^{2} y

$2 x^{2} y$

y^{2} x

$y^{2} x$

0

$0$

9 x^{4}

$9 x^{4}$

3 x^{3} y

$3 x^{3} y$

5 x^{2} y^{2}

$5 x^{2} y^{2}$

y^{3} x

$y^{3} x$

0

$0$

9 x^{4}

$9 x^{4}$

6 x^{3} y

$6 x^{3} y$

3 x^{2} y^{2}

$3 x^{2} y^{2}$

0

$0$

3 x^{3} y

$3 x^{3} y$

2 x^{2} y^{2}

$2 x^{2} y^{2}$

y^{3} x

$y^{3} x$

Step 6

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

3 x

$3 x$

3 x^{3}

$3 x^{3}$

2 x^{2} y

$2 x^{2} y$

y^{2} x

$y^{2} x$

0

$0$

9 x^{4}

$9 x^{4}$

3 x^{3} y

$3 x^{3} y$

5 x^{2} y^{2}

$5 x^{2} y^{2}$

y^{3} x

$y^{3} x$

0

$0$

9 x^{4}

$9 x^{4}$

6 x^{3} y

$6 x^{3} y$

3 x^{2} y^{2}

$3 x^{2} y^{2}$

0

$0$

3 x^{3} y

$3 x^{3} y$

2 x^{2} y^{2}

$2 x^{2} y^{2}$

y^{3} x

$y^{3} x$

0

$0$

Step 7

Divide the highest order term in the dividend

- 3 x^{3} y

$- 3 x^{3} y$ by the highest order term in divisor

3 x^{3}

$3 x^{3}$ .

3 x

$3 x$

y

$y$

3 x^{3}

$3 x^{3}$

2 x^{2} y

$2 x^{2} y$

y^{2} x

$y^{2} x$

0

$0$

9 x^{4}

$9 x^{4}$

$3 x^{3} y$

$5 x^{2} y^{2}$

$y^{3} x$

$0$

$9 x^{4}$

$6 x^{3} y$

$3 x^{2} y^{2}$

$0$

$3 x^{3} y$

$2 x^{2} y^{2}$

$y^{3} x$

$0$

Step 8

Multiply the new quotient term by the divisor.

$3 x$

$y$

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

$5 x^{2} y^{2}$

$y^{3} x$

$0$

$9 x^{4}$

$6 x^{3} y$

$3 x^{2} y^{2}$

$0$

$3 x^{3} y$

$2 x^{2} y^{2}$

$y^{3} x$

$0$

$3 y x^{3}$

$2 y^{2} x^{2}$

$y^{3} x$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in

$- 3 y x^{3} - 2 y^{2} x^{2} + y^{3} x + 0$

$3 x$

$y$

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

$5 x^{2} y^{2}$

$y^{3} x$

$0$

$9 x^{4}$

$6 x^{3} y$

$3 x^{2} y^{2}$

$0$

$3 x^{3} y$

$2 x^{2} y^{2}$

$y^{3} x$

$0$

$3 y x^{3}$

$2 y^{2} x^{2}$

$y^{3} x$

$0$

Step 10

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

$3 x$

$y$

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

$5 x^{2} y^{2}$

$y^{3} x$

$0$

$9 x^{4}$

$6 x^{3} y$

$3 x^{2} y^{2}$

$0$

$3 x^{3} y$

$2 x^{2} y^{2}$

$y^{3} x$

$0$

$3 y x^{3}$

$2 y^{2} x^{2}$

$y^{3} x$

$0$

Step 11

Since the remander is

$0$ , the final answer is the quotient.

$3 x - y$