Algebra Examples

$6 x^{3} + 2 x^{2} + 18 x + 26$ by $x^{2} + 3$

Step 1

Write the problem as a mathematical expression.

$\frac{6 x^{3} + 2 x^{2} + 18 x + 26}{x^{2} + 3}$

Step 2

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

$x^{2}$

$0 x$

$3$

$6 x^{3}$

$2 x^{2}$

$18 x$

$26$

Step 3

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

$6 x$

$x^{2}$

$0 x$

$3$

$6 x^{3}$

$2 x^{2}$

$18 x$

$26$

Step 4

Multiply the new quotient term by the divisor.

$6 x$

$x^{2}$

$0 x$

$3$

$6 x^{3}$

$2 x^{2}$

$18 x$

$26$

$6 x^{3}$

$0$

$18 x$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} + 0 + 18 x$

$6 x$

$x^{2}$

$0 x$

$3$

$6 x^{3}$

$2 x^{2}$

$18 x$

$26$

$6 x^{3}$

$0$

$18 x$

Step 6

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

$6 x$

$x^{2}$

$0 x$

$3$

$6 x^{3}$

$2 x^{2}$

$18 x$

$26$

$6 x^{3}$

$0$

$18 x$

$2 x^{2}$

$0$

Step 7

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

$6 x$

$x^{2}$

$0 x$

$3$

$6 x^{3}$

$2 x^{2}$

$18 x$

$26$

$6 x^{3}$

$0$

$18 x$

$2 x^{2}$

$0$

$26$

Step 8

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

$6 x$

$2$

$x^{2}$

$0 x$

$3$

$6 x^{3}$

$2 x^{2}$

$18 x$

$26$

$6 x^{3}$

$0$

$18 x$

$2 x^{2}$

$0$

$26$

Step 9

Multiply the new quotient term by the divisor.

$6 x$

$2$

$x^{2}$

$0 x$

$3$

$6 x^{3}$

$2 x^{2}$

$18 x$

$26$

$6 x^{3}$

$0$

$18 x$

$2 x^{2}$

$0$

$26$

$2 x^{2}$

$0$

$6$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} + 0 + 6$

						$6 x$	+	$2$
$x^{2}$	+	$0 x$	+	$3$		$6 x^{3}$	+	$2 x^{2}$	+	$18 x$	+	$26$
					-	$6 x^{3}$	-	$0$	-	$18 x$
							+	$2 x^{2}$	+	$0$	+	$26$
							-	$2 x^{2}$	-	$0$	-	$6$

Step 11

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

						$6 x$	+	$2$
$x^{2}$	+	$0 x$	+	$3$		$6 x^{3}$	+	$2 x^{2}$	+	$18 x$	+	$26$
					-	$6 x^{3}$	-	$0$	-	$18 x$
							+	$2 x^{2}$	+	$0$	+	$26$
							-	$2 x^{2}$	-	$0$	-	$6$
											+	$20$

Step 12

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

$6 x + 2 + \frac{20}{x^{2} + 3}$