Algebra Examples

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

Step 1

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

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

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

$y^{3} x$

$0$

Step 2

Divide the highest order term in the dividend $9 x^{4}$ by the highest order term in divisor $3 x^{3}$ .

$3 x$

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

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

$y^{3} x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$3 x$

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

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

$y^{3} x$

$0$

$9 x^{4}$

$6 x^{3} y$

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

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{4} + 6 x^{3} y - 3 x^{2} y^{2} + 0$

$3 x$

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

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

$y^{3} x$

$0$

$9 x^{4}$

$6 x^{3} y$

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

$0$

Step 5

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

$3 x$

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

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

$y^{3} x$

$0$

$9 x^{4}$

$6 x^{3} y$

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

$0$

$3 x^{3} y$

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

$y^{3} x$

Step 6

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

$3 x$

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

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

$y^{3} x$

$0$

$9 x^{4}$

$6 x^{3} y$

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

$0$

$3 x^{3} y$

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

$y^{3} x$

$0$

Step 7

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

$3 x$

$y$

$3 x^{3}$

$2 x^{2} y$

$y^{2} x$

$0$

$9 x^{4}$

$3 x^{3} y$

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

$y^{3} x$

$0$

$9 x^{4}$

$6 x^{3} y$

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

$0$

$3 x^{3} y$

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

$y^{3} x$

$0$

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.

$3 x - y$