Divide Using Long Polynomial Division (4x^5-x^3-2x^2-180x+14)/(x^2-7)

$\frac{4 x^{5} - x^{3} - 2 x^{2} - 180 x + 14}{x^{2} - 7}$

Step 1

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

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

Step 2

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

$4 x^{3}$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

Step 3

Multiply the new quotient term by the divisor.

$4 x^{3}$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

Step 4

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

$4 x^{3}$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

Step 5

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

$4 x^{3}$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

$27 x^{3}$

Step 6

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

$4 x^{3}$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

$27 x^{3}$

$2 x^{2}$

$180 x$

Step 7

Divide the highest order term in the dividend $27 x^{3}$ by the highest order term in divisor $x^{2}$ .

$4 x^{3}$

$0 x^{2}$

$27 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

$27 x^{3}$

$2 x^{2}$

$180 x$

Step 8

Multiply the new quotient term by the divisor.

$4 x^{3}$

$0 x^{2}$

$27 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

$27 x^{3}$

$2 x^{2}$

$180 x$

$27 x^{3}$

$0$

$189 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $27 x^{3} + 0 - 189 x$

$4 x^{3}$

$0 x^{2}$

$27 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

$27 x^{3}$

$2 x^{2}$

$180 x$

$27 x^{3}$

$0$

$189 x$

Step 10

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

$4 x^{3}$

$0 x^{2}$

$27 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

$27 x^{3}$

$2 x^{2}$

$180 x$

$27 x^{3}$

$0$

$189 x$

$2 x^{2}$

$9 x$

Step 11

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

$4 x^{3}$

$0 x^{2}$

$27 x$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

$27 x^{3}$

$2 x^{2}$

$180 x$

$27 x^{3}$

$0$

$189 x$

$2 x^{2}$

$9 x$

$14$

Step 12

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

$4 x^{3}$

$0 x^{2}$

$27 x$

$2$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

$27 x^{3}$

$2 x^{2}$

$180 x$

$27 x^{3}$

$0$

$189 x$

$2 x^{2}$

$9 x$

$14$

Step 13

Multiply the new quotient term by the divisor.

$4 x^{3}$

$0 x^{2}$

$27 x$

$2$

$x^{2}$

$0 x$

$7$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$2 x^{2}$

$180 x$

$14$

$4 x^{5}$

$0$

$28 x^{3}$

$27 x^{3}$

$2 x^{2}$

$180 x$

$27 x^{3}$

$0$

$189 x$

$2 x^{2}$

$9 x$

$14$

$2 x^{2}$

$0$

$14$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} + 0 + 14$

						$4 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$2$
$x^{2}$	+	$0 x$	-	$7$		$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	-	$2 x^{2}$	-	$180 x$	+	$14$
					-	$4 x^{5}$	-	$0$	+	$28 x^{3}$
									+	$27 x^{3}$	-	$2 x^{2}$	-	$180 x$
									-	$27 x^{3}$	-	$0$	+	$189 x$
											-	$2 x^{2}$	+	$9 x$	+	$14$
											+	$2 x^{2}$	-	$0$	-	$14$

Step 15

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

						$4 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$2$
$x^{2}$	+	$0 x$	-	$7$		$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	-	$2 x^{2}$	-	$180 x$	+	$14$
					-	$4 x^{5}$	-	$0$	+	$28 x^{3}$
									+	$27 x^{3}$	-	$2 x^{2}$	-	$180 x$
									-	$27 x^{3}$	-	$0$	+	$189 x$
											-	$2 x^{2}$	+	$9 x$	+	$14$
											+	$2 x^{2}$	-	$0$	-	$14$
													+	$9 x$	+	$0$

Step 16

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

$4 x^{3} + 27 x - 2 + \frac{9 x}{x^{2} - 7}$

Divide Using Long Polynomial Division (4x^5-x^3-2x^2-180x+14)/(x^2-7)

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