Algebra Examples

$(a^{4} + 4 b^{4}) \div (a^{2} - 2 a b + 2 b^{2})$

Step 1

Move $- 2 a b$ .

$\frac{a^{4} + 4 b^{4}}{a^{2} + 2 b^{2} - 2 a b}$

Step 2

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

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

Step 3

Divide the highest order term in the dividend $a^{4}$ by the highest order term in divisor $a^{2}$ .

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

Step 4

Multiply the new quotient term by the divisor.

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

Step 5

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

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

Step 6

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

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

Step 7

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

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

Step 8

Divide the highest order term in the dividend $2 a^{3} b$ by the highest order term in divisor $a^{2}$ .

$a^{2}$

$2 a b$

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

Step 9

Multiply the new quotient term by the divisor.

$a^{2}$

$2 a b$

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

$2 a^{3} b$

$4 a^{2} b^{2}$

$4 a b^{3}$

Step 10

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

$a^{2}$

$2 a b$

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

$2 a^{3} b$

$4 a^{2} b^{2}$

$4 a b^{3}$

Step 11

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

$a^{2}$

$2 a b$

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

$2 a^{3} b$

$4 a^{2} b^{2}$

$4 a b^{3}$

$2 a^{2} b^{2}$

$4 a b^{3}$

Step 12

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

$a^{2}$

$2 a b$

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

$2 a^{3} b$

$4 a^{2} b^{2}$

$4 a b^{3}$

$2 a^{2} b^{2}$

$4 a b^{3}$

$4 b^{4}$

Step 13

Divide the highest order term in the dividend $2 a^{2} b^{2}$ by the highest order term in divisor $a^{2}$ .

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

$2 a^{3} b$

$4 a^{2} b^{2}$

$4 a b^{3}$

$2 a^{2} b^{2}$

$4 a b^{3}$

$4 b^{4}$

Step 14

Multiply the new quotient term by the divisor.

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

$2 a^{3} b$

$4 a^{2} b^{2}$

$4 a b^{3}$

$2 a^{2} b^{2}$

$4 a b^{3}$

$4 b^{4}$

$2 b^{2} a^{2}$

$4 b^{3} a$

$4 b^{4}$

Step 15

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

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

$2 a^{3} b$

$4 a^{2} b^{2}$

$4 a b^{3}$

$2 a^{2} b^{2}$

$4 a b^{3}$

$4 b^{4}$

$2 b^{2} a^{2}$

$4 b^{3} a$

$4 b^{4}$

Step 16

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

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{2}$

$2 a b$

$2 b^{2}$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$4 b^{4}$

$a^{4}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$2 a^{3} b$

$2 a^{2} b^{2}$

$0 a$

$2 a^{3} b$

$4 a^{2} b^{2}$

$4 a b^{3}$

$2 a^{2} b^{2}$

$4 a b^{3}$

$4 b^{4}$

$2 b^{2} a^{2}$

$4 b^{3} a$

$4 b^{4}$

$0$

Step 17

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

$a^{2} + 2 a b + 2 b^{2}$