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Chapter 5: Problem 82

Write an equivalent expression by factoring. $$3 a^{2}+6 a+30+7 a^{2} b+14 a b+70 b$$

Short Answer

Expert verified
(a^2 + 2a + 10)(3 + 7b)

Step by step solution

01

Group the Terms

First, group the terms into two sets: \( 3a^2 + 6a + 30 \) and \( 7a^2b + 14ab + 70b \).
02

Factor Common Factors

Factor out the greatest common factor (GCF) from each group: For the first group: \( 3(a^2 + 2a + 10) \) For the second group: \( 7b(a^2 + 2a + 10) \)
03

Combine the Factored Expressions

Notice that both factored expressions have a common binomial \( (a^2 + 2a + 10) \): \( 3(a^2 + 2a + 10) + 7b(a^2 + 2a + 10) \)
04

Factor Out the Common Binomial

Factor out the common binomial \( (a^2 + 2a + 10) \) from the entire expression:\((a^2 + 2a + 10)(3 + 7b)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Grouping Terms
When starting to factor a polynomial, one of the initial steps involves grouping terms. Look at all the terms and identify how they can be grouped into smaller expressions. This often involves creating pairs or sets of terms that have something in common.
The original problem breaks down into two sets of terms: [3a^2 + 6a + 30] and [7a^2b + 14ab + 70b] by grouping terms with common factors together.
Grouping terms makes it easier to apply further factoring techniques.
Greatest Common Factor (GCF)
After grouping terms, the next step is to find the Greatest Common Factor (GCF) for each group.
The GCF is the largest expression that can evenly divide each term in the group.
For the first group [3a^2 + 6a + 30], the GCF is 3. This means each term in the group can be divided by 3. So, we factor out 3:
3(a^2 + 2a + 10).
For the second group [7a^2b + 14ab + 70b], the GCF is 7b. That allows us to factor out 7b:
7b(a^2 + 2a + 10).
Understanding the GCF helps simplify the expressions and reveals common patterns.
Factored Expressions
Once the terms are grouped and the common factors are factored out, you might notice that the resulting expressions have something in common.
From our problem, we have: 3(a^2 + 2a + 10) for the first group and 7b(a^2 + 2a + 10) for the second group.
Both these expressions share the common binomial: (a^2 + 2a + 10).
You can then factor out this common binomial:
(a^2 + 2a + 10)(3 + 7b).
The factored expressions represent the original polynomial in a more simplified and comprehensible form.
This method of factoring is essential for solving complex polynomials.

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Most popular questions from this chapter

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