Algebra Examples

$\frac{9 x^{5} + 3 x^{4} - 27 x^{3} - 3 x^{2} + 18 x}{3 x^{3} - 3 x}$

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

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

Divide the highest order term in the dividend $9 x^{5}$ by the highest order term in divisor $3 x^{3}$ .

$3 x^{2}$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

Multiply the new quotient term by the divisor.

$3 x^{2}$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{5} + 0 - 9 x^{3} + 0$

$3 x^{2}$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

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

$3 x^{2}$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

$3 x^{4}$

$18 x^{3}$

$3 x^{2}$

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

$3 x^{2}$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

$3 x^{4}$

$18 x^{3}$

$3 x^{2}$

$18 x$

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

$3 x^{2}$

$x$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

$3 x^{4}$

$18 x^{3}$

$3 x^{2}$

$18 x$

Multiply the new quotient term by the divisor.

$3 x^{2}$

$x$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

$3 x^{4}$

$18 x^{3}$

$3 x^{2}$

$18 x$

$3 x^{4}$

$0$

$3 x^{2}$

$0$

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} + 0 - 3 x^{2} + 0$

$3 x^{2}$

$x$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

$3 x^{4}$

$18 x^{3}$

$3 x^{2}$

$18 x$

$3 x^{4}$

$0$

$3 x^{2}$

$0$

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

$3 x^{2}$

$x$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

$3 x^{4}$

$18 x^{3}$

$3 x^{2}$

$18 x$

$3 x^{4}$

$0$

$3 x^{2}$

$0$

$18 x^{3}$

$0$

$18 x$

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

$3 x^{2}$

$x$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

$3 x^{4}$

$18 x^{3}$

$3 x^{2}$

$18 x$

$3 x^{4}$

$0$

$3 x^{2}$

$0$

$18 x^{3}$

$0$

$18 x$

$0$

Divide the highest order term in the dividend $- 18 x^{3}$ by the highest order term in divisor $3 x^{3}$ .

								$3 x^{2}$	+	$x$	-	$6$
$3 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$0$		$9 x^{5}$	+	$3 x^{4}$	-	$27 x^{3}$	-	$3 x^{2}$	+	$18 x$	+	$0$
							-	$9 x^{5}$	-	$0$	+	$9 x^{3}$	-	$0$
									+	$3 x^{4}$	-	$18 x^{3}$	-	$3 x^{2}$	+	$18 x$
									-	$3 x^{4}$	-	$0$	+	$3 x^{2}$	-	$0$
											-	$18 x^{3}$	+	$0$	+	$18 x$	+	$0$

Multiply the new quotient term by the divisor.

$3 x^{2}$

$x$

$6$

$3 x^{3}$

$0 x^{2}$

$3 x$

$0$

$9 x^{5}$

$3 x^{4}$

$27 x^{3}$

$3 x^{2}$

$18 x$

$0$

$9 x^{5}$

$0$

$9 x^{3}$

$0$

$3 x^{4}$

$18 x^{3}$

$3 x^{2}$

$18 x$

$3 x^{4}$

$0$

$3 x^{2}$

$0$

$18 x^{3}$

$0$

$18 x$

$0$

$18 x^{3}$

$0$

$18 x$

$0$

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

								$3 x^{2}$	+	$x$	-	$6$
$3 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$0$		$9 x^{5}$	+	$3 x^{4}$	-	$27 x^{3}$	-	$3 x^{2}$	+	$18 x$	+	$0$
							-	$9 x^{5}$	-	$0$	+	$9 x^{3}$	-	$0$
									+	$3 x^{4}$	-	$18 x^{3}$	-	$3 x^{2}$	+	$18 x$
									-	$3 x^{4}$	-	$0$	+	$3 x^{2}$	-	$0$
											-	$18 x^{3}$	+	$0$	+	$18 x$	+	$0$
											+	$18 x^{3}$	-	$0$	-	$18 x$	-	$0$

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

								$3 x^{2}$	+	$x$	-	$6$
$3 x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$0$		$9 x^{5}$	+	$3 x^{4}$	-	$27 x^{3}$	-	$3 x^{2}$	+	$18 x$	+	$0$
							-	$9 x^{5}$	-	$0$	+	$9 x^{3}$	-	$0$
									+	$3 x^{4}$	-	$18 x^{3}$	-	$3 x^{2}$	+	$18 x$
									-	$3 x^{4}$	-	$0$	+	$3 x^{2}$	-	$0$
											-	$18 x^{3}$	+	$0$	+	$18 x$	+	$0$
											+	$18 x^{3}$	-	$0$	-	$18 x$	-	$0$
																		$0$

Since the remander is $0$ , the final answer is the quotient.

$3 x^{2} + x - 6$