Divide Using Synthetic Division (2x^3-x^2-18x+32)/(2x-6)

$\frac{2 x^{3} - x^{2} - 18 x + 32}{2 x - 6}$

Step 1

Divide each term in the denominator by $2$ to make the coefficient of linear factor variable $1$ .

$\frac{1}{2} \cdot \frac{2 x^{3} - x^{2} - 18 x + 32}{x - 3}$

Step 2

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

$3$	$2$	$- 1$	$- 18$	$32$
	\xa0			\xa0

Step 3

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$3$	$2$	$- 1$	$- 18$	$32$
	\xa0			\xa0
	$2$

Step 4

Multiply the newest entry in the result $(2)$ by the divisor $(3)$ and place the result of $(6)$ under the next term in the dividend $(- 1)$ .

$3$	$2$	$- 1$	$- 18$	$32$
	\xa0	$6$		\xa0
	$2$

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.

$3$	$2$	$- 1$	$- 18$	$32$
	\xa0	$6$		\xa0
	$2$	$5$

Step 6

Multiply the newest entry in the result $(5)$ by the divisor $(3)$ and place the result of $(15)$ under the next term in the dividend $(- 18)$ .

$3$	$2$	$- 1$	$- 18$	$32$
	\xa0	$6$	$15$	\xa0
	$2$	$5$

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.

$3$	$2$	$- 1$	$- 18$	$32$
	\xa0	$6$	$15$	\xa0
	$2$	$5$	$- 3$

Step 8

Multiply the newest entry in the result $(- 3)$ by the divisor $(3)$ and place the result of $(- 9)$ under the next term in the dividend $(32)$ .

$3$	$2$	$- 1$	$- 18$	$32$
	\xa0	$6$	$15$	$- 9$
	$2$	$5$	$- 3$

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.

$3$	$2$	$- 1$	$- 18$	$32$
	\xa0	$6$	$15$	$- 9$
	$2$	$5$	$- 3$	$23$

Step 10

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

$(\frac{1}{2}) \cdot (2 x^{2} + 5 x - 3 + \frac{23}{x - 3})$

Step 11

Simplify.

Distribute.

Apply the distributive property.

$\frac{1}{2} \cdot (2 x^{2} + 5 x - 3) + \frac{1}{2} \cdot \frac{23}{x - 3}$

Apply the distributive property.

$\frac{1}{2} \cdot (2 x^{2} + 5 x) + \frac{1}{2} \cdot - 3 + \frac{1}{2} \cdot \frac{23}{x - 3}$

Apply the distributive property.

$\frac{1}{2} \cdot (2 x^{2}) + \frac{1}{2} \cdot (5 x) + \frac{1}{2} \cdot - 3 + \frac{1}{2} \cdot \frac{23}{x - 3}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x^{2}$ .

$\frac{1}{2} \cdot (2 (x^{2})) + \frac{1}{2} \cdot (5 x) + \frac{1}{2} \cdot - 3 + \frac{1}{2} \cdot \frac{23}{x - 3}$

Cancel the common factor.

$\frac{1}{2} \cdot (2 x^{2}) + \frac{1}{2} \cdot (5 x) + \frac{1}{2} \cdot - 3 + \frac{1}{2} \cdot \frac{23}{x - 3}$

Rewrite the expression.

$x^{2} + \frac{1}{2} \cdot (5 x) + \frac{1}{2} \cdot - 3 + \frac{1}{2} \cdot \frac{23}{x - 3}$

Multiply $\frac{1}{2} (5 x)$ .

Combine $5$ and $\frac{1}{2}$ .

$x^{2} + \frac{5}{2} x + \frac{1}{2} \cdot - 3 + \frac{1}{2} \cdot \frac{23}{x - 3}$

Combine $\frac{5}{2}$ and $x$ .

$x^{2} + \frac{5 x}{2} + \frac{1}{2} \cdot - 3 + \frac{1}{2} \cdot \frac{23}{x - 3}$

Combine $\frac{1}{2}$ and $- 3$ .

$x^{2} + \frac{5 x}{2} + \frac{- 3}{2} + \frac{1}{2} \cdot \frac{23}{x - 3}$

Move the negative in front of the fraction.

$x^{2} + \frac{5 x}{2} - \frac{3}{2} + \frac{1}{2} \cdot \frac{23}{x - 3}$

Multiply $\frac{1}{2}$ by $\frac{23}{x - 3}$ .

$x^{2} + \frac{5 x}{2} - \frac{3}{2} + \frac{23}{2 (x - 3)}$

Divide Using Synthetic Division (2x^3-x^2-18x+32)/(2x-6)

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