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Chapter 5: Problem 54

Factor completely. $$ 2 b-28 a^{2} b+10 a b $$

Short Answer

Expert verified
The factored form is \(2b(1 - 14a^2 + 5a)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, find the GCF of all the terms in the expression. The terms are: \(2b\), \(-28a^2b\), and \(10ab\). The common factor in all terms is \(2b\).
02

Factor out the GCF

Factor out \(2b\) from each term of the expression. This gives us: \[2b(1 - 14a^2 + 5a)\]
03

Simplify the expression inside the parenthesis

Now, we look at the expression inside the parenthesis: \(1 - 14a^2 + 5a\). Check if it can be factored further. In this case, it cannot be simplified further efficiently.

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Key Concepts

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

greatest common factor
When factoring algebraic expressions, the first step is to find the greatest common factor (GCF). This means finding the largest number or expression that divides all terms in your expression. It simplifies the expression and makes further factoring easier. In the problem given, the terms are: \(2b\), \(-28a^2b\), and \(10ab\). We look for the GCF of the coefficients (2, -28, and 10) and the variables they share (b). The GCF here is \(2b\), as it divides evenly into each term. Factoring out \(2b\) results in: \(2b(1 - 14a^2 + 5a)\).
factoring binomials
Factoring binomials involves expressing them as a product of simpler binomials or monomials. Although the original problem is a trinomial, understanding binomials helps in more complex expressions. For instance, the expression inside the parenthesis \(1 - 14a^2 + 5a\) could sometimes be factored as a binomial, which is useful in breaking it down further. Binomials like \(x^2 - y^2\) can be factored into \((x+y)(x-y)\). In our exercise, however, \(1 - 14a^2 + 5a\) does not factor further neatly, but this knowledge prepares you for similar problems.
intermediate algebra
Intermediate algebra prepares students for more advanced algebra concepts and involves techniques like factoring. Here, we simplify expressions, solve quadratic equations, or manipulate polynomials. In the exercise, after factoring out the GCF, we have \(2b(1 - 14a^2 + 5a)\). An intermediate algebra skill is recognizing if the remaining expression can be factored further. Even though \(1 - 14a^2 + 5a\) cannot be simplified more efficiently, understanding these steps is crucial. Intermediate algebra often involves breaking down complex problems into simpler parts, a skill showcased perfectly in the given example.

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