Q 6.52 - Intermediate Algebra

Question

Factor: $27p^{2}q+90pq+75q$.

Step-by-Step Solution

Answer

Factors of $27p^{2}q+90pq+75q$ are $3q{\left(3p+5\right)}^2$.

Step 1. Given information.

Given an expression $27p^{2}q+90pq+75q$.

Step 2. Find if GCF of all the three terms.

GCD stands for greatest common factor.

Here, $3q$ is GCF of $27p^{2}q+90pq+75q$.

So, the expression can be written as $3q\left(9p^2+30p+25\right)$.

Step 3. Find if it is a perfect trinomial.

A perfect trinomial is of the form $a^2\pm 2ab+b^2$. .

Consider $a=3p,b=5$.

Following result can be obtained.

 $\begin{array}{rcl}3q\left(9p^2+30p+25\right)& =& 3q\left({\left(3q\right)}^2+2\left(3p\right)\left(5\right)+5^2\right)\\ & =& 3q\left(a^2+2ab+b^2\right)\end{array}$

Therefore, the expression is a perfect trinomial and can be expressed as $3q{\left(3p+5\right)}^2$.

Step 4. Verify the result.

Multiply the factors as follows.

$\begin{array}{rcl}3q{\left(3p+5\right)}^2& =& 3q\left({\left(3p\right)}^2+2\left(3p\right)\left(5\right)+5^2\right)\\ & =& 3q\left(9p^2+30p+25\right)\\ & =& 27p^{2}q+90pq+75q\end{array}$