Find the Remainder (7w^4+8w^2-1)÷(w-1/7)

$(7 w^{4} + 8 w^{2} - 1) \div (w - \frac{1}{7})$

Step 1

To calculate the remainder, first divide the polynomials.

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

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

Divide the highest order term in the dividend $7 w^{4}$ by the highest order term in divisor $w$ .

$7 w^{3}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

Multiply the new quotient term by the divisor.

$7 w^{3}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $7 w^{4} - w^{3}$

$7 w^{3}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

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

$7 w^{3}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

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

$7 w^{3}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

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

$7 w^{3}$

$w^{2}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

Multiply the new quotient term by the divisor.

$7 w^{3}$

$w^{2}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

The expression needs to be subtracted from the dividend, so change all the signs in $w^{3} - \frac{w^{2}}{7}$

$7 w^{3}$

$w^{2}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

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

$7 w^{3}$

$w^{2}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

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

$7 w^{3}$

$w^{2}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

Divide the highest order term in the dividend $\frac{57 w^{2}}{7}$ by the highest order term in divisor $w$ .

$7 w^{3}$

$w^{2}$

$\frac{57 w}{7}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

Multiply the new quotient term by the divisor.

$7 w^{3}$

$w^{2}$

$\frac{57 w}{7}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

$\frac{57 w^{2}}{7}$

$\frac{57 w}{49}$

The expression needs to be subtracted from the dividend, so change all the signs in $\frac{57 w^{2}}{7} - \frac{57 w}{49}$

$7 w^{3}$

$w^{2}$

$\frac{57 w}{7}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

$\frac{57 w^{2}}{7}$

$\frac{57 w}{49}$

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

$7 w^{3}$

$w^{2}$

$\frac{57 w}{7}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

$\frac{57 w^{2}}{7}$

$\frac{57 w}{49}$

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

$7 w^{3}$

$w^{2}$

$\frac{57 w}{7}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

$\frac{57 w^{2}}{7}$

$\frac{57 w}{49}$

$1$

Divide the highest order term in the dividend $\frac{57 w}{49}$ by the highest order term in divisor $w$ .

$7 w^{3}$

$w^{2}$

$\frac{57 w}{7}$

$\frac{57}{49}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

$\frac{57 w^{2}}{7}$

$\frac{57 w}{49}$

$1$

Multiply the new quotient term by the divisor.

$7 w^{3}$

$w^{2}$

$\frac{57 w}{7}$

$\frac{57}{49}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

$\frac{57 w^{2}}{7}$

$\frac{57 w}{49}$

$1$

$\frac{57 w}{49}$

$\frac{57}{343}$

The expression needs to be subtracted from the dividend, so change all the signs in $\frac{57 w}{49} - \frac{57}{343}$

$7 w^{3}$

$w^{2}$

$\frac{57 w}{7}$

$\frac{57}{49}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

$\frac{57 w^{2}}{7}$

$\frac{57 w}{49}$

$1$

$\frac{57 w}{49}$

$\frac{57}{343}$

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

$7 w^{3}$

$w^{2}$

$\frac{57 w}{7}$

$\frac{57}{49}$

$w$

$\frac{1}{7}$

$7 w^{4}$

$0 w^{3}$

$8 w^{2}$

$0 w$

$1$

$7 w^{4}$

$w^{3}$

$8 w^{2}$

$w^{3}$

$\frac{w^{2}}{7}$

$\frac{57 w^{2}}{7}$

$0 w$

$\frac{57 w^{2}}{7}$

$\frac{57 w}{49}$

$1$

$\frac{57 w}{49}$

$\frac{57}{343}$

$\frac{286}{343}$

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

$7 w^{3} + w^{2} + \frac{57 w}{7} + \frac{57}{49} - \frac{286}{343 (w - \frac{1}{7})}$

Step 2

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

$- 286$

Find the Remainder (7w^4+8w^2-1)÷(w-1/7)

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