Divide Using Synthetic Division (6x^5+5x^4+x^3-3x^2+x)/(3x+1)

$\frac{6 x^{5} + 5 x^{4} + x^{3} - 3 x^{2} + x}{3 x + 1}$

Step 1

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

$\frac{1}{3} \cdot \frac{6 x^{5} + 5 x^{4} + x^{3} - 3 x^{2} + x}{\frac{3 x + 1}{3}}$

Step 2

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

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0					\xa0

Step 3

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0					\xa0
	$6$

Step 4

Multiply the newest entry in the result $(6)$ by the divisor $(- \frac{1}{3})$ and place the result of $(- 2)$ under the next term in the dividend $(5)$ .

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$				\xa0
	$6$

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.

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$				\xa0
	$6$	$3$

Step 6

Multiply the newest entry in the result $(3)$ by the divisor $(- \frac{1}{3})$ and place the result of $(- 1)$ under the next term in the dividend $(1)$ .

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$	$- 1$			\xa0
	$6$	$3$

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.

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$	$- 1$			\xa0
	$6$	$3$	$0$

Step 8

Multiply the newest entry in the result $(0)$ by the divisor $(- \frac{1}{3})$ and place the result of $(0)$ under the next term in the dividend $(- 3)$ .

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$	$- 1$	$0$		\xa0
	$6$	$3$	$0$

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.

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$	$- 1$	$0$		\xa0
	$6$	$3$	$0$	$- 3$

Step 10

Multiply the newest entry in the result $(- 3)$ by the divisor $(- \frac{1}{3})$ and place the result of $(1)$ under the next term in the dividend $(1)$ .

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$	$- 1$	$0$	$1$	\xa0
	$6$	$3$	$0$	$- 3$

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.

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$	$- 1$	$0$	$1$	\xa0
	$6$	$3$	$0$	$- 3$	$2$

Step 12

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{1}{3})$ and place the result of $(- \frac{2}{3})$ under the next term in the dividend $(0)$ .

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$	$- 1$	$0$	$1$	$- \frac{2}{3}$
	$6$	$3$	$0$	$- 3$	$2$

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.

$- \frac{1}{3}$	$6$	$5$	$1$	$- 3$	$1$	$0$
	\xa0	$- 2$	$- 1$	$0$	$1$	$- \frac{2}{3}$
	$6$	$3$	$0$	$- 3$	$2$	$- \frac{2}{3}$

Step 14

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

$(\frac{1}{3}) \cdot (6 x^{4} + 3 x^{3} + 0 x^{2} + - 3 x + 2 + \frac{- \frac{2}{3}}{\frac{3 x + 1}{3}})$

Step 15

Simplify the quotient polynomial.

$(\frac{1}{3}) \cdot (6 x^{4} + 3 x^{3} - 3 x + 2 - \frac{2}{3 x + 1})$

Step 16

Simplify.

Distribute.

Apply the distributive property.

$\frac{1}{3} \cdot (6 x^{4} + 3 x^{3} - 3 x + 2) + \frac{1}{3} \cdot (- \frac{2}{3 x + 1})$

Apply the distributive property.

$\frac{1}{3} \cdot (6 x^{4} + 3 x^{3} - 3 x) + \frac{1}{3} \cdot 2 + \frac{1}{3} \cdot (- \frac{2}{3 x + 1})$

Apply the distributive property.

$\frac{1}{3} \cdot (6 x^{4} + 3 x^{3}) + \frac{1}{3} \cdot (- 3 x) + \frac{1}{3} \cdot 2 + \frac{1}{3} \cdot (- \frac{2}{3 x + 1})$

Apply the distributive property.

$\frac{1}{3} \cdot (6 x^{4}) + \frac{1}{3} \cdot (3 x^{3}) + \frac{1}{3} \cdot (- 3 x) + \frac{1}{3} \cdot 2 + \frac{1}{3} \cdot (- \frac{2}{3 x + 1})$

$\frac{1}{3} \cdot (6 x^{4}) + \frac{1}{3} \cdot (3 x^{3}) + \frac{1}{3} \cdot (- 3 x) + \frac{1}{3} \cdot 2 + \frac{1}{3} \cdot (- \frac{2}{3 x + 1})$

Cancel the common factor of $3$ .

Factor $3$ out of $6 x^{4}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 x^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor of $3$ .

Factor $3$ out of $- 3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

$2 x^{4} + x^{3} - x + \frac{2}{3} - \frac{2}{3 (3 x + 1)}$

$2 x^{4} + x^{3} - x + \frac{2}{3} - \frac{2}{3 (3 x + 1)}$

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