Find the Remainder (9x^4+10x^2-1)÷(x-1/9)

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

To calculate the remainder, first divide the polynomials.

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

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

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

$9 x^{3}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

Multiply the new quotient term by the divisor.

$9 x^{3}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

$9 x^{4}$

–

$x^{3}$

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

$9 x^{3}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

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

$9 x^{3}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

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

$9 x^{3}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

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

$9 x^{3}$

$x^{2}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

Multiply the new quotient term by the divisor.

$9 x^{3}$

$x^{2}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

$x^{3}$

–

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

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

$9 x^{3}$

$x^{2}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$9 x^{3}$

$x^{2}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

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

$9 x^{3}$

$x^{2}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

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

$9 x^{3}$

$x^{2}$

$\frac{91 x}{9}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

Multiply the new quotient term by the divisor.

$9 x^{3}$

$x^{2}$

$\frac{91 x}{9}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

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

–

$\frac{91 x}{81}$

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

$9 x^{3}$

$x^{2}$

$\frac{91 x}{9}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

–

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

$\frac{91 x}{81}$

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

$9 x^{3}$

$x^{2}$

$\frac{91 x}{9}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

–

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

$\frac{91 x}{81}$

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

$9 x^{3}$

$x^{2}$

$\frac{91 x}{9}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

–

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

$\frac{91 x}{81}$

–

$1$

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

$9 x^{3}$

$x^{2}$

$\frac{91 x}{9}$

$\frac{91}{81}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

–

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

$\frac{91 x}{81}$

–

$1$

Multiply the new quotient term by the divisor.

$9 x^{3}$

$x^{2}$

$\frac{91 x}{9}$

$\frac{91}{81}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

–

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

$\frac{91 x}{81}$

–

$1$

$\frac{91 x}{81}$

–

$\frac{91}{729}$

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

$9 x^{3}$

$x^{2}$

$\frac{91 x}{9}$

$\frac{91}{81}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

–

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

$\frac{91 x}{81}$

–

$1$

–

$\frac{91 x}{81}$

$\frac{91}{729}$

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

$9 x^{3}$

$x^{2}$

$\frac{91 x}{9}$

$\frac{91}{81}$

$x$

–

$\frac{1}{9}$

$9 x^{4}$

$0 x^{3}$

$10 x^{2}$

$0 x$

–

$1$

–

$9 x^{4}$

$x^{3}$

$10 x^{2}$

–

$x^{3}$

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

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

$0 x$

–

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

$\frac{91 x}{81}$

–

$1$

–

$\frac{91 x}{81}$

$\frac{91}{729}$

–

$\frac{638}{729}$

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

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

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

$- 638$

Find the Remainder (9x^4+10x^2-1)÷(x-1/9)

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