Factor:$3p^3-6p^{2}q+9p{q}^3$

The given expression is$3p^3-6p^{2}q+9p{q}^3$.

• Factor each of the terms in the given expression.

$$\begin{gathered} 3p^3=3·p·p·p \\ 6p^{2}q=3·2·p·p·q \\ 9p{q}^3=3·3·p·q·q \end{gathered}$$

• From the obtained factors, it is observed that the common factors are $3,p$.
• Multiply the common factors to determine the GCF of the terms of the given expression.

$3·p=3p$

• So, the GCF of the terms of the given expression is $3p$.

• Express each of the term of the given expression as a product of GCF, $3p$.

$3p^3-6p^{2}q+9p{q}^3=3p·p^2-3p·2pq+3p·3q^3$

• Use the reverse distributive property, $ab+ac=a(b+c)$to rewrite the obtained expression.

$3p^2(3p-2pq+3q^3)$

• Multiply the obtained factors.

$\begin{array}{rcl}3p^2(3p-2pq+3q^3)& =& 3p^2·3p-3p^2·2pq+3p^2·3q^3\\ & =& 9p^3-6p^{3}q+9p^2{q}^3\end{array}$

• Since the obtained expression is the same as the given expression, the obtained factors are verified.