Divide Using Long Polynomial Division (25a^4+19a^2-a-78)÷(5a^2+6a-9)

$(25 a^{4} + 19 a^{2} - a - 78) \div (5 a^{2} + 6 a - 9)$

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

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

Divide the highest order term in the dividend $25 a^{4}$ by the highest order term in divisor $5 a^{2}$ .

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

Multiply the new quotient term by the divisor.

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

$25 a^{4}$

$30 a^{3}$

–

$45 a^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $25 a^{4} + 30 a^{3} - 45 a^{2}$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

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

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

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

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

Divide the highest order term in the dividend $- 30 a^{3}$ by the highest order term in divisor $5 a^{2}$ .

$5 a^{2}$

–

$6 a$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

Multiply the new quotient term by the divisor.

$5 a^{2}$

–

$6 a$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

–

$30 a^{3}$

–

$36 a^{2}$

$54 a$

The expression needs to be subtracted from the dividend, so change all the signs in $- 30 a^{3} - 36 a^{2} + 54 a$

$5 a^{2}$

–

$6 a$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

$30 a^{3}$

$36 a^{2}$

–

$54 a$

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

$5 a^{2}$

–

$6 a$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

$30 a^{3}$

$36 a^{2}$

–

$54 a$

$100 a^{2}$

–

$55 a$

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

$5 a^{2}$

–

$6 a$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

$30 a^{3}$

$36 a^{2}$

–

$54 a$

$100 a^{2}$

–

$55 a$

–

$78$

Divide the highest order term in the dividend $100 a^{2}$ by the highest order term in divisor $5 a^{2}$ .

$5 a^{2}$

–

$6 a$

$20$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

$30 a^{3}$

$36 a^{2}$

–

$54 a$

$100 a^{2}$

–

$55 a$

–

$78$

Multiply the new quotient term by the divisor.

$5 a^{2}$

–

$6 a$

$20$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

$30 a^{3}$

$36 a^{2}$

–

$54 a$

$100 a^{2}$

–

$55 a$

–

$78$

$100 a^{2}$

$120 a$

–

$180$

The expression needs to be subtracted from the dividend, so change all the signs in $100 a^{2} + 120 a - 180$

$5 a^{2}$

–

$6 a$

$20$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

$30 a^{3}$

$36 a^{2}$

–

$54 a$

$100 a^{2}$

–

$55 a$

–

$78$

–

$100 a^{2}$

–

$120 a$

$180$

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

$5 a^{2}$

–

$6 a$

$20$

$5 a^{2}$

$6 a$

–

$9$

$25 a^{4}$

$0 a^{3}$

$19 a^{2}$

–

$a$

–

$78$

–

$25 a^{4}$

–

$30 a^{3}$

$45 a^{2}$

–

$30 a^{3}$

$64 a^{2}$

–

$a$

$30 a^{3}$

$36 a^{2}$

–

$54 a$

$100 a^{2}$

–

$55 a$

–

$78$

–

$100 a^{2}$

–

$120 a$

$180$

–

$175 a$

$102$

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

$5 a^{2} - 6 a + 20 + \frac{- 175 a + 102}{5 a^{2} + 6 a - 9}$

Divide Using Long Polynomial Division (25a^4+19a^2-a-78)÷(5a^2+6a-9)

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App from the store