Algebra Examples

$(6 x^{3} + x^{2} - 10 x - 1) \div (3 x^{2} - 4 x - 2)$

Step 1

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

$3 x^{2}$

$4 x$

$2$

$6 x^{3}$

$x^{2}$

$10 x$

$1$

Step 2

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $3 x^{2}$ .

							$2 x$
	$3 x^{2}$	-	$4 x$	-	$2$		$6 x^{3}$	+	$x^{2}$	-	$10 x$	-	$1$

Step 3

Multiply the new quotient term by the divisor.

						$2 x$
$3 x^{2}$	-	$4 x$	-	$2$		$6 x^{3}$	+	$x^{2}$	-	$10 x$	-	$1$
					+	$6 x^{3}$	-	$8 x^{2}$	-	$4 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} - 8 x^{2} - 4 x$

$2 x$

$3 x^{2}$

$4 x$

$2$

$6 x^{3}$

$x^{2}$

$10 x$

$1$

$6 x^{3}$

$8 x^{2}$

$4 x$

Step 5

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

$2 x$

$3 x^{2}$

$4 x$

$2$

$6 x^{3}$

$x^{2}$

$10 x$

$1$

$6 x^{3}$

$8 x^{2}$

$4 x$

$9 x^{2}$

$6 x$

Step 6

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

						$2 x$
$3 x^{2}$	-	$4 x$	-	$2$		$6 x^{3}$	+	$x^{2}$	-	$10 x$	-	$1$
					-	$6 x^{3}$	+	$8 x^{2}$	+	$4 x$
							+	$9 x^{2}$	-	$6 x$	-	$1$

Step 7

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $3 x^{2}$ .

						$2 x$	+	$3$
$3 x^{2}$	-	$4 x$	-	$2$		$6 x^{3}$	+	$x^{2}$	-	$10 x$	-	$1$
					-	$6 x^{3}$	+	$8 x^{2}$	+	$4 x$
							+	$9 x^{2}$	-	$6 x$	-	$1$

Step 8

Multiply the new quotient term by the divisor.

						$2 x$	+	$3$
$3 x^{2}$	-	$4 x$	-	$2$		$6 x^{3}$	+	$x^{2}$	-	$10 x$	-	$1$
					-	$6 x^{3}$	+	$8 x^{2}$	+	$4 x$
							+	$9 x^{2}$	-	$6 x$	-	$1$
							+	$9 x^{2}$	-	$12 x$	-	$6$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} - 12 x - 6$

$2 x$

$3$

$3 x^{2}$

$4 x$

$2$

$6 x^{3}$

$x^{2}$

$10 x$

$1$

$6 x^{3}$

$8 x^{2}$

$4 x$

$9 x^{2}$

$6 x$

$1$

$9 x^{2}$

$12 x$

$6$

Step 10

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

$2 x$

$3$

$3 x^{2}$

$4 x$

$2$

$6 x^{3}$

$x^{2}$

$10 x$

$1$

$6 x^{3}$

$8 x^{2}$

$4 x$

$9 x^{2}$

$6 x$

$1$

$9 x^{2}$

$12 x$

$6$

$6 x$

$5$

Step 11

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

$2 x + 3 + \frac{6 x + 5}{3 x^{2} - 4 x - 2}$