Divide Using Synthetic Division (4p^3-p^2+2p)÷(3p-1)

$(4 p^{3} - p^{2} + 2 p) \div (3 p - 1)$

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

$\frac{1}{3} \cdot \frac{4 p^{3} - p^{2} + 2 p}{\frac{3 p - 1}{3}}$

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

$\frac{1}{3}$	$4$	$- 1$	$2$	$0$
	\xa0			\xa0

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$\frac{1}{3}$	$4$	$- 1$	$2$	$0$
	\xa0			\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{1}{3})$ and place the result of $(\frac{4}{3})$ under the next term in the dividend $(- 1)$ .

$\frac{1}{3}$	$4$	$- 1$	$2$	$0$
	\xa0	$\frac{4}{3}$		\xa0
	$4$

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

$\frac{1}{3}$	$4$	$- 1$	$2$	$0$
	\xa0	$\frac{4}{3}$		\xa0
	$4$	$\frac{1}{3}$

Multiply the newest entry in the result $(\frac{1}{3})$ by the divisor $(\frac{1}{3})$ and place the result of $(\frac{1}{9})$ under the next term in the dividend $(2)$ .

$\frac{1}{3}$	$4$	$- 1$	$2$	$0$
	\xa0	$\frac{4}{3}$	$\frac{1}{9}$	\xa0
	$4$	$\frac{1}{3}$

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

$\frac{1}{3}$	$4$	$- 1$	$2$	$0$
	\xa0	$\frac{4}{3}$	$\frac{1}{9}$	\xa0
	$4$	$\frac{1}{3}$	$\frac{19}{9}$

Multiply the newest entry in the result $(\frac{19}{9})$ by the divisor $(\frac{1}{3})$ and place the result of $(\frac{19}{27})$ under the next term in the dividend $(0)$ .

$\frac{1}{3}$	$4$	$- 1$	$2$	$0$
	\xa0	$\frac{4}{3}$	$\frac{1}{9}$	$\frac{19}{27}$
	$4$	$\frac{1}{3}$	$\frac{19}{9}$

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

$\frac{1}{3}$	$4$	$- 1$	$2$	$0$
	\xa0	$\frac{4}{3}$	$\frac{1}{9}$	$\frac{19}{27}$
	$4$	$\frac{1}{3}$	$\frac{19}{9}$	$\frac{19}{27}$

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

$(\frac{1}{3}) \cdot (4 p^{2} + \frac{1}{3} p + \frac{19}{9} + \frac{\frac{19}{27}}{\frac{3 p - 1}{3}})$

Simplify the quotient polynomial.

$(\frac{1}{3}) \cdot (4 p^{2} + \frac{p}{3} + \frac{19}{9} + \frac{19}{9 (3 p - 1)})$

Simplify.

Distribute.

Apply the distributive property.

$\frac{1}{3} \cdot (4 p^{2} + \frac{p}{3} + \frac{19}{9} + \frac{19}{9 (3 p) + 9 \cdot - 1})$

Apply the distributive property.

$\frac{1}{3} \cdot (4 p^{2} + \frac{p}{3} + \frac{19}{9}) + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 \cdot - 1}$

Apply the distributive property.

$\frac{1}{3} \cdot (4 p^{2} + \frac{p}{3}) + \frac{1}{3} \cdot \frac{19}{9} + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 \cdot - 1}$

Apply the distributive property.

$\frac{1}{3} \cdot (4 p^{2}) + \frac{1}{3} \cdot \frac{p}{3} + \frac{1}{3} \cdot \frac{19}{9} + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 \cdot - 1}$

Multiply $\frac{1}{3} (4 p^{2})$ .

Combine $4$ and $\frac{1}{3}$ .

$\frac{4}{3} p^{2} + \frac{1}{3} \cdot \frac{p}{3} + \frac{1}{3} \cdot \frac{19}{9} + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 \cdot - 1}$

Combine $\frac{4}{3}$ and $p^{2}$ .

$\frac{4 p^{2}}{3} + \frac{1}{3} \cdot \frac{p}{3} + \frac{1}{3} \cdot \frac{19}{9} + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 \cdot - 1}$

Multiply $\frac{1}{3} \cdot \frac{p}{3}$ .

Multiply $\frac{1}{3}$ by $\frac{p}{3}$ .

$\frac{4 p^{2}}{3} + \frac{p}{3 \cdot 3} + \frac{1}{3} \cdot \frac{19}{9} + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 \cdot - 1}$

Multiply $3$ by $3$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{1}{3} \cdot \frac{19}{9} + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 \cdot - 1}$

Multiply $\frac{1}{3} \cdot \frac{19}{9}$ .

Multiply $\frac{1}{3}$ by $\frac{19}{9}$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{19}{3 \cdot 9} + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 \cdot - 1}$

Multiply $3$ by $9$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{19}{27} + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 \cdot - 1}$

Simplify the denominator.

Multiply $3$ by $9$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{19}{27} + \frac{1}{3} \cdot \frac{19}{27 p + 9 \cdot - 1}$

Multiply $9$ by $- 1$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{19}{27} + \frac{1}{3} \cdot \frac{19}{27 p - 9}$

Factor $9$ out of $27 p - 9$ .

Factor $9$ out of $27 p$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{19}{27} + \frac{1}{3} \cdot \frac{19}{9 (3 p) - 9}$

Factor $9$ out of $- 9$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{19}{27} + \frac{1}{3} \cdot \frac{19}{9 (3 p) + 9 (- 1)}$

Factor $9$ out of $9 (3 p) + 9 (- 1)$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{19}{27} + \frac{1}{3} \cdot \frac{19}{9 (3 p - 1)}$

Multiply $\frac{1}{3} \cdot \frac{19}{9 (3 p - 1)}$ .

Multiply $\frac{1}{3}$ by $\frac{19}{9 (3 p - 1)}$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{19}{27} + \frac{19}{3 (9 (3 p - 1))}$

Multiply $9$ by $3$ .

$\frac{4 p^{2}}{3} + \frac{p}{9} + \frac{19}{27} + \frac{19}{27 (3 p - 1)}$

Divide Using Synthetic Division (4p^3-p^2+2p)÷(3p-1)

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