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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 6: Q 6.52 (page 595)

Factor: $27 p^{2} q + 90 p q + 75 q$ .

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

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

02

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

GCD stands for greatest common factor.

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

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

03

Step 3. Find if it is a perfect trinomial.

A perfect trinomial is of the form $a^{2} 卤 2 a b + b^{2}$ ..

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

Following result can be obtained.

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

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

04

Step 4. Verify the result.

Multiply the factors as follows.

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

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Most popular questions from this chapter

Factor by Grouping In the following exercises, factor by grouping.

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