Algebra Examples

$\frac{16 a^{4} - 40 a^{2} + 24 a}{12 a^{3}}$

Step 1

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

$12 a^{3}$

$0 a^{2}$

$0 a$

$0$

$16 a^{4}$

$0 a^{3}$

$40 a^{2}$

$24 a$

$0$

Step 2

Divide the highest order term in the dividend $16 a^{4}$ by the highest order term in divisor $12 a^{3}$ .

$\frac{4 a}{3}$

$12 a^{3}$

$0 a^{2}$

$0 a$

$0$

$16 a^{4}$

$0 a^{3}$

$40 a^{2}$

$24 a$

$0$

Step 3

Multiply the new quotient term by the divisor.

$\frac{4 a}{3}$

$12 a^{3}$

$0 a^{2}$

$0 a$

$0$

$16 a^{4}$

$0 a^{3}$

$40 a^{2}$

$24 a$

$0$

$16 a^{4}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $16 a^{4} + 0 + 0 + 0$

								$\frac{4 a}{3}$
$12 a^{3}$	+	$0 a^{2}$	+	$0 a$	+	$0$		$16 a^{4}$	+	$0 a^{3}$	-	$40 a^{2}$	+	$24 a$	+	$0$
							-	$16 a^{4}$	-	$0$	-	$0$	-	$0$

Step 5

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

$\frac{4 a}{3}$

$12 a^{3}$

$0 a^{2}$

$0 a$

$0$

$16 a^{4}$

$0 a^{3}$

$40 a^{2}$

$24 a$

$0$

$16 a^{4}$

$0$

$40 a^{2}$

$24 a$

Step 6

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

$\frac{4 a}{3}$

$12 a^{3}$

$0 a^{2}$

$0 a$

$0$

$16 a^{4}$

$0 a^{3}$

$40 a^{2}$

$24 a$

$0$

$16 a^{4}$

$0$

$40 a^{2}$

$24 a$

$0$

Step 7

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

$\frac{4 a}{3} + \frac{- 40 a^{2} + 24 a}{12 a^{3}}$