Algebra Examples

$(9 w^{4} + 10 w^{2} - 1) \div (w - \frac{1}{9})$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

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

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

Step 1.2

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

$9 w^{3}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

Step 1.3

Multiply the new quotient term by the divisor.

$9 w^{3}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $9 w^{4} - w^{3}$

$9 w^{3}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

Step 1.5

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

$9 w^{3}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

Step 1.6

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

$9 w^{3}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

Step 1.7

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

$9 w^{3}$

$w^{2}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

Step 1.8

Multiply the new quotient term by the divisor.

$9 w^{3}$

$w^{2}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $w^{3} - \frac{w^{2}}{9}$

$9 w^{3}$

$w^{2}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

Step 1.10

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

$9 w^{3}$

$w^{2}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

Step 1.11

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

$9 w^{3}$

$w^{2}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

Step 1.12

Divide the highest order term in the dividend $\frac{91 w^{2}}{9}$ by the highest order term in divisor $w$ .

$9 w^{3}$

$w^{2}$

$\frac{91 w}{9}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

Step 1.13

Multiply the new quotient term by the divisor.

$9 w^{3}$

$w^{2}$

$\frac{91 w}{9}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

$\frac{91 w^{2}}{9}$

$\frac{91 w}{81}$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $\frac{91 w^{2}}{9} - \frac{91 w}{81}$

$9 w^{3}$

$w^{2}$

$\frac{91 w}{9}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

$\frac{91 w^{2}}{9}$

$\frac{91 w}{81}$

Step 1.15

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

$9 w^{3}$

$w^{2}$

$\frac{91 w}{9}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

$\frac{91 w^{2}}{9}$

$\frac{91 w}{81}$

Step 1.16

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

$9 w^{3}$

$w^{2}$

$\frac{91 w}{9}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

$\frac{91 w^{2}}{9}$

$\frac{91 w}{81}$

$1$

Step 1.17

Divide the highest order term in the dividend $\frac{91 w}{81}$ by the highest order term in divisor $w$ .

$9 w^{3}$

$w^{2}$

$\frac{91 w}{9}$

$\frac{91}{81}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

$\frac{91 w^{2}}{9}$

$\frac{91 w}{81}$

$1$

Step 1.18

Multiply the new quotient term by the divisor.

$9 w^{3}$

$w^{2}$

$\frac{91 w}{9}$

$\frac{91}{81}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

$\frac{91 w^{2}}{9}$

$\frac{91 w}{81}$

$1$

$\frac{91 w}{81}$

$\frac{91}{729}$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $\frac{91 w}{81} - \frac{91}{729}$

$9 w^{3}$

$w^{2}$

$\frac{91 w}{9}$

$\frac{91}{81}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

$\frac{91 w^{2}}{9}$

$\frac{91 w}{81}$

$1$

$\frac{91 w}{81}$

$\frac{91}{729}$

Step 1.20

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

$9 w^{3}$

$w^{2}$

$\frac{91 w}{9}$

$\frac{91}{81}$

$w$

$\frac{1}{9}$

$9 w^{4}$

$0 w^{3}$

$10 w^{2}$

$0 w$

$1$

$9 w^{4}$

$w^{3}$

$10 w^{2}$

$w^{3}$

$\frac{w^{2}}{9}$

$\frac{91 w^{2}}{9}$

$0 w$

$\frac{91 w^{2}}{9}$

$\frac{91 w}{81}$

$1$

$\frac{91 w}{81}$

$\frac{91}{729}$

$\frac{638}{729}$

Step 1.21

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

$9 w^{3} + w^{2} + \frac{91 w}{9} + \frac{91}{81} - \frac{638}{729 (w - \frac{1}{9})}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$- 638$