Find the Remainder (h^4+h^2-2)÷(h+3)

$(h^{4} + h^{2} - 2) \div (h + 3)$

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 $0$ .

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

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

$h^{3}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

Multiply the new quotient term by the divisor.

$h^{3}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

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

$h^{3}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

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

$h^{3}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

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

$h^{3}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

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

$h^{3}$

$3 h^{2}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

Multiply the new quotient term by the divisor.

$h^{3}$

$3 h^{2}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

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

$h^{3}$

$3 h^{2}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

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

$h^{3}$

$3 h^{2}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

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

$h^{3}$

$3 h^{2}$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

Divide the highest order term in the dividend $10 h^{2}$ by the highest order term in divisor $h$ .

$h^{3}$

$3 h^{2}$

$10 h$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

Multiply the new quotient term by the divisor.

$h^{3}$

$3 h^{2}$

$10 h$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

$10 h^{2}$

$30 h$

The expression needs to be subtracted from the dividend, so change all the signs in $10 h^{2} + 30 h$

$h^{3}$

$3 h^{2}$

$10 h$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

$10 h^{2}$

$30 h$

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

$h^{3}$

$3 h^{2}$

$10 h$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

$10 h^{2}$

$30 h$

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

$h^{3}$

$3 h^{2}$

$10 h$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

$10 h^{2}$

$30 h$

$2$

Divide the highest order term in the dividend $- 30 h$ by the highest order term in divisor $h$ .

$h^{3}$

$3 h^{2}$

$10 h$

$30$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

$10 h^{2}$

$30 h$

$2$

Multiply the new quotient term by the divisor.

$h^{3}$

$3 h^{2}$

$10 h$

$30$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

$10 h^{2}$

$30 h$

$2$

$30 h$

$90$

The expression needs to be subtracted from the dividend, so change all the signs in $- 30 h - 90$

$h^{3}$

$3 h^{2}$

$10 h$

$30$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

$10 h^{2}$

$30 h$

$2$

$30 h$

$90$

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

$h^{3}$

$3 h^{2}$

$10 h$

$30$

$h$

$3$

$h^{4}$

$0 h^{3}$

$h^{2}$

$0 h$

$2$

$h^{4}$

$3 h^{3}$

$h^{2}$

$3 h^{3}$

$9 h^{2}$

$10 h^{2}$

$0 h$

$10 h^{2}$

$30 h$

$2$

$30 h$

$90$

$88$

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

$h^{3} - 3 h^{2} + 10 h - 30 + \frac{88}{h + 3}$

Step 2

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

$88$

Find the Remainder (h^4+h^2-2)÷(h+3)

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