Divide Using Synthetic Division (-15x^5+10x^4+15x^2-37x+18)÷(3x-2)

$(- 15 x^{5} + 10 x^{4} + 15 x^{2} - 37 x + 18) \div (3 x - 2)$

Step 1

Divide each term in the denominator by $3$ to make the coefficient of linear factor variable $1$ .

$\frac{1}{3} \cdot \frac{- 15 x^{5} + 10 x^{4} + 15 x^{2} - 37 x + 18}{\frac{3 x - 2}{3}}$

Step 2

Place the numbers representing the divisor and the dividend into a division-like configuration.

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0					\xa0

Step 3

The first number in the dividend $(- 15)$ is put into the first position of the result area (below the horizontal line).

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0					\xa0
	$- 15$

Step 4

Multiply the newest entry in the result $(- 15)$ by the divisor $(\frac{2}{3})$ and place the result of $(- 10)$ under the next term in the dividend $(10)$ .

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$				\xa0
	$- 15$

Step 5

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$				\xa0
	$- 15$	$0$

Step 6

Multiply the newest entry in the result $(0)$ by the divisor $(\frac{2}{3})$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$	$0$			\xa0
	$- 15$	$0$

Step 7

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$	$0$			\xa0
	$- 15$	$0$	$0$

Step 8

Multiply the newest entry in the result $(0)$ by the divisor $(\frac{2}{3})$ and place the result of $(0)$ under the next term in the dividend $(15)$ .

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$	$0$	$0$		\xa0
	$- 15$	$0$	$0$

Step 9

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$	$0$	$0$		\xa0
	$- 15$	$0$	$0$	$15$

Step 10

Multiply the newest entry in the result $(15)$ by the divisor $(\frac{2}{3})$ and place the result of $(10)$ under the next term in the dividend $(- 37)$ .

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$	$0$	$0$	$10$	\xa0
	$- 15$	$0$	$0$	$15$

Step 11

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$	$0$	$0$	$10$	\xa0
	$- 15$	$0$	$0$	$15$	$- 27$

Step 12

Multiply the newest entry in the result $(- 27)$ by the divisor $(\frac{2}{3})$ and place the result of $(- 18)$ under the next term in the dividend $(18)$ .

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$	$0$	$0$	$10$	$- 18$
	$- 15$	$0$	$0$	$15$	$- 27$

Step 13

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{2}{3}$	$- 15$	$10$	$0$	$15$	$- 37$	$18$
	\xa0	$- 10$	$0$	$0$	$10$	$- 18$
	$- 15$	$0$	$0$	$15$	$- 27$	$0$

Step 14

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$(\frac{1}{3}) \cdot (- 15 x^{4} + 0 x^{3} + 0 x^{2} + 15 x - 27)$

Step 15

Simplify the quotient polynomial.

$(\frac{1}{3}) \cdot (- 15 x^{4} + 15 x - 27)$

Step 16

Simplify.

Distribute.

Apply the distributive property.

$\frac{1}{3} \cdot (- 15 x^{4} + 15 x) + \frac{1}{3} \cdot - 27$

Apply the distributive property.

$\frac{1}{3} \cdot (- 15 x^{4}) + \frac{1}{3} \cdot (15 x) + \frac{1}{3} \cdot - 27$

$\frac{1}{3} \cdot (- 15 x^{4}) + \frac{1}{3} \cdot (15 x) + \frac{1}{3} \cdot - 27$

Cancel the common factor of $3$ .

Factor $3$ out of $- 15 x^{4}$ .

$\frac{1}{3} \cdot (3 (- 5 x^{4})) + \frac{1}{3} \cdot (15 x) + \frac{1}{3} \cdot - 27$

Cancel the common factor.

$\frac{1}{3} \cdot (3 (- 5 x^{4})) + \frac{1}{3} \cdot (15 x) + \frac{1}{3} \cdot - 27$

Rewrite the expression.

$- 5 x^{4} + \frac{1}{3} \cdot (15 x) + \frac{1}{3} \cdot - 27$

$- 5 x^{4} + \frac{1}{3} \cdot (15 x) + \frac{1}{3} \cdot - 27$

Cancel the common factor of $3$ .

Factor $3$ out of $15 x$ .

$- 5 x^{4} + \frac{1}{3} \cdot (3 (5 x)) + \frac{1}{3} \cdot - 27$

Cancel the common factor.

$- 5 x^{4} + \frac{1}{3} \cdot (3 (5 x)) + \frac{1}{3} \cdot - 27$

Rewrite the expression.

$- 5 x^{4} + 5 x + \frac{1}{3} \cdot - 27$

$- 5 x^{4} + 5 x + \frac{1}{3} \cdot - 27$

Cancel the common factor of $3$ .

Factor $3$ out of $- 27$ .

$- 5 x^{4} + 5 x + \frac{1}{3} \cdot (3 (- 9))$

Cancel the common factor.

$- 5 x^{4} + 5 x + \frac{1}{3} \cdot (3 \cdot - 9)$

Rewrite the expression.

$- 5 x^{4} + 5 x - 9$

$- 5 x^{4} + 5 x - 9$

$- 5 x^{4} + 5 x - 9$

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