Algebra Examples

$(5 x^{4} - 32 x^{3} - 20 x^{2} - 13 x + 42) \div (7 - x)$

Step 1

Reorder $7$ and $- x$ .

$\frac{5 x^{4} - 32 x^{3} - 20 x^{2} - 13 x + 42}{- x + 7}$

Step 2

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

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

Step 3

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

				-	$5 x^{3}$
-	$x$	+	$7$		$5 x^{4}$	-	$32 x^{3}$	-	$20 x^{2}$	-	$13 x$	+	$42$

Step 4

Multiply the new quotient term by the divisor.

				-	$5 x^{3}$
-	$x$	+	$7$		$5 x^{4}$	-	$32 x^{3}$	-	$20 x^{2}$	-	$13 x$	+	$42$
				+	$5 x^{4}$	-	$35 x^{3}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} - 35 x^{3}$

				-	$5 x^{3}$
-	$x$	+	$7$		$5 x^{4}$	-	$32 x^{3}$	-	$20 x^{2}$	-	$13 x$	+	$42$
				-	$5 x^{4}$	+	$35 x^{3}$

Step 6

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

$5 x^{3}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

Step 7

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

$5 x^{3}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

Step 8

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

				-	$5 x^{3}$	-	$3 x^{2}$
-	$x$	+	$7$		$5 x^{4}$	-	$32 x^{3}$	-	$20 x^{2}$	-	$13 x$	+	$42$
				-	$5 x^{4}$	+	$35 x^{3}$
						+	$3 x^{3}$	-	$20 x^{2}$

Step 9

Multiply the new quotient term by the divisor.

$5 x^{3}$

$3 x^{2}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 21 x^{2}$

$5 x^{3}$

$3 x^{2}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

Step 11

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

$5 x^{3}$

$3 x^{2}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

Step 12

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

$5 x^{3}$

$3 x^{2}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

$13 x$

Step 13

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

$5 x^{3}$

$3 x^{2}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

$13 x$

Step 14

Multiply the new quotient term by the divisor.

$5 x^{3}$

$3 x^{2}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

$13 x$

$x^{2}$

$7 x$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} - 7 x$

$5 x^{3}$

$3 x^{2}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

$13 x$

$x^{2}$

$7 x$

Step 16

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

$5 x^{3}$

$3 x^{2}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

$13 x$

$x^{2}$

$7 x$

$6 x$

Step 17

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

$5 x^{3}$

$3 x^{2}$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

$13 x$

$x^{2}$

$7 x$

$6 x$

$42$

Step 18

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $- x$ .

$5 x^{3}$

$3 x^{2}$

$x$

$6$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

$13 x$

$x^{2}$

$7 x$

$6 x$

$42$

Step 19

Multiply the new quotient term by the divisor.

$5 x^{3}$

$3 x^{2}$

$x$

$6$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

$13 x$

$x^{2}$

$7 x$

$6 x$

$42$

$6 x$

$42$

Step 20

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x + 42$

$5 x^{3}$

$3 x^{2}$

$x$

$6$

$x$

$7$

$5 x^{4}$

$32 x^{3}$

$20 x^{2}$

$13 x$

$42$

$5 x^{4}$

$35 x^{3}$

$3 x^{3}$

$20 x^{2}$

$3 x^{3}$

$21 x^{2}$

$x^{2}$

$13 x$

$x^{2}$

$7 x$

$6 x$

$42$

$6 x$

$42$

Step 21

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

				-	$5 x^{3}$	-	$3 x^{2}$	-	$x$	+	$6$
-	$x$	+	$7$		$5 x^{4}$	-	$32 x^{3}$	-	$20 x^{2}$	-	$13 x$	+	$42$
				-	$5 x^{4}$	+	$35 x^{3}$
						+	$3 x^{3}$	-	$20 x^{2}$
						-	$3 x^{3}$	+	$21 x^{2}$
								+	$x^{2}$	-	$13 x$
								-	$x^{2}$	+	$7 x$
										-	$6 x$	+	$42$
										+	$6 x$	-	$42$
													$0$

Step 22

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

$- 5 x^{3} - 3 x^{2} - x + 6$