Chapter 5.2, Problem 34WE

To determine

To find: The missing factor $18p{q}^3=\left(6pq\right)(?)$

Answer to Problem 34WE

The missing factor is $3q^2$ .

Explanation of Solution

Given information:

The polynomial’s factorization is $18p{q}^3=\left(6pq\right)(?)$

Formula used:

If ${\text{P}}_{\text{1}}$ , ${\text{P}}_{\text{2}}$ , and ${\text{P}}_{\text{3}}$ are three polynomial such that ${\text{P}}_{\text{1}}\text{=}{\text{P}}_{\text{2}}\text{×}{\text{P}}_{\text{3}}$ , then $\frac{{\text{P}}_{\text{1}}}{{\text{P}}_{\text{2}}}\text{=}{\text{P}}_{\text{3}}$ , where ${\text{P}}_{\text{2}}$ , and ${\text{P}}_{\text{3}}$ are factors of ${\text{P}}_{\text{1}}$ .

Calculations:

The polynomial’s factorization is $18p{q}^3=\left(6pq\right)(?)$ . Let missing factor is x , then $18p{q}^3=\left(6pq\right)\times x$

  $\Rightarrow x=\frac{18p{q}^3}{6pq}$

  $\Rightarrow x=\frac{18p{q}^3}{6pq}=\frac{\cancel{2}\times \cancel{3}\times 3\times \cancel{p}\times \cancel{q}\times q\times q}{\cancel{2}\times \cancel{3}\times \cancel{p}\times \cancel{q}}$

  $\Rightarrow x=3\times q\times q=3q^2$

Thus, the missing factor is $3q^2$

Conclusion:

The missing factor is $3q^2$ .