Precalculus Examples

$\frac{- 4 x^{5} - x^{3} - 5 x^{2} + 212 x + 35}{x^{2} - 7}$

Step 1

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

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$212 x$

$35$

Step 2

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

$4 x^{3}$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$212 x$

$35$

Step 3

Multiply the new quotient term by the divisor.

$4 x^{3}$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$212 x$

$35$

$4 x^{5}$

$0$

$28 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{5} + 0 + 28 x^{3}$

					-	$4 x^{3}$
$x^{2}$	+	$0 x$	-	$7$	-	$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	-	$5 x^{2}$	+	$212 x$	+	$35$
					+	$4 x^{5}$	-	$0$	-	$28 x^{3}$

Step 5

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

					-	$4 x^{3}$
$x^{2}$	+	$0 x$	-	$7$	-	$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	-	$5 x^{2}$	+	$212 x$	+	$35$
					+	$4 x^{5}$	-	$0$	-	$28 x^{3}$
									-	$29 x^{3}$

Step 6

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

$4 x^{3}$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$212 x$

$35$

$4 x^{5}$

$0$

$28 x^{3}$

$29 x^{3}$

$5 x^{2}$

$212 x$

Step 7

Divide the highest order term in the dividend $- 29 x^{3}$ by the highest order term in divisor $x^{2}$ .

$4 x^{3}$

$0 x^{2}$

$29 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$212 x$

$35$

$4 x^{5}$

$0$

$28 x^{3}$

$29 x^{3}$

$5 x^{2}$

$212 x$

Step 8

Multiply the new quotient term by the divisor.

$4 x^{3}$

$0 x^{2}$

$29 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$212 x$

$35$

$4 x^{5}$

$0$

$28 x^{3}$

$29 x^{3}$

$5 x^{2}$

$212 x$

$29 x^{3}$

$0$

$203 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 29 x^{3} + 0 + 203 x$

$4 x^{3}$

$0 x^{2}$

$29 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$212 x$

$35$

$4 x^{5}$

$0$

$28 x^{3}$

$29 x^{3}$

$5 x^{2}$

$212 x$

$29 x^{3}$

$0$

$203 x$

Step 10

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

$4 x^{3}$

$0 x^{2}$

$29 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$212 x$

$35$

$4 x^{5}$

$0$

$28 x^{3}$

$29 x^{3}$

$5 x^{2}$

$212 x$

$29 x^{3}$

$0$

$203 x$

$5 x^{2}$

$9 x$

Step 11

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

$4 x^{3}$

$0 x^{2}$

$29 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$212 x$

$35$

$4 x^{5}$

$0$

$28 x^{3}$

$29 x^{3}$

$5 x^{2}$

$212 x$

$29 x^{3}$

$0$

$203 x$

$5 x^{2}$

$9 x$

$35$

Step 12

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

					-	$4 x^{3}$	+	$0 x^{2}$	-	$29 x$	-	$5$
$x^{2}$	+	$0 x$	-	$7$	-	$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	-	$5 x^{2}$	+	$212 x$	+	$35$
					+	$4 x^{5}$	-	$0$	-	$28 x^{3}$
									-	$29 x^{3}$	-	$5 x^{2}$	+	$212 x$
									+	$29 x^{3}$	-	$0$	-	$203 x$
											-	$5 x^{2}$	+	$9 x$	+	$35$

Step 13

Multiply the new quotient term by the divisor.

					-	$4 x^{3}$	+	$0 x^{2}$	-	$29 x$	-	$5$
$x^{2}$	+	$0 x$	-	$7$	-	$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	-	$5 x^{2}$	+	$212 x$	+	$35$
					+	$4 x^{5}$	-	$0$	-	$28 x^{3}$
									-	$29 x^{3}$	-	$5 x^{2}$	+	$212 x$
									+	$29 x^{3}$	-	$0$	-	$203 x$
											-	$5 x^{2}$	+	$9 x$	+	$35$
											-	$5 x^{2}$	+	$0$	+	$35$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} + 0 + 35$

					-	$4 x^{3}$	+	$0 x^{2}$	-	$29 x$	-	$5$
$x^{2}$	+	$0 x$	-	$7$	-	$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	-	$5 x^{2}$	+	$212 x$	+	$35$
					+	$4 x^{5}$	-	$0$	-	$28 x^{3}$
									-	$29 x^{3}$	-	$5 x^{2}$	+	$212 x$
									+	$29 x^{3}$	-	$0$	-	$203 x$
											-	$5 x^{2}$	+	$9 x$	+	$35$
											+	$5 x^{2}$	-	$0$	-	$35$

Step 15

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

					-	$4 x^{3}$	+	$0 x^{2}$	-	$29 x$	-	$5$
$x^{2}$	+	$0 x$	-	$7$	-	$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	-	$5 x^{2}$	+	$212 x$	+	$35$
					+	$4 x^{5}$	-	$0$	-	$28 x^{3}$
									-	$29 x^{3}$	-	$5 x^{2}$	+	$212 x$
									+	$29 x^{3}$	-	$0$	-	$203 x$
											-	$5 x^{2}$	+	$9 x$	+	$35$
											+	$5 x^{2}$	-	$0$	-	$35$
													+	$9 x$	+	$0$

Step 16

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

$- 4 x^{3} - 29 x - 5 + \frac{9 x}{x^{2} - 7}$