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Chapter 7: Problem 49

Factor by grouping. \(b^{2}+5 b-4 b-20\)

Short Answer

Expert verified
(b + 5)(b - 4)

Step by step solution

01

Group the terms

Group the terms into two pairs: you have the polynomial: \[b^2 + 5b - 4b - 20\]. Group the terms as follows: \[(b^2 + 5b) + (-4b - 20)\].
02

Factor out the greatest common factor (GCF) from each pair

For the first group \[b^2 + 5b\],factor out the GCF, which is \(b\): \[b(b + 5)\].For the second group \[-4b - 20\],factor out the GCF, which is \(-4\): \[-4(b + 5)\].
03

Factor out the common binomial factor

Notice that both terms now contain a common binomial factor \[(b + 5)\], so factor \(b + 5\) out of the expression: \[b(b + 5) - 4(b + 5)\] simplifying to \[(b + 5)(b - 4)\].

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Key Concepts

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

factor by grouping
Factoring by grouping is an effective method to factor polynomials that have four or more terms. When we look at the polynomial expression in the original problem: Step 1: We first group the terms into pairs to make the expression easier to work with. Instead of looking at the whole polynomial Step 2: We look at grouped pairs Step 3: Factor out the greatest common factor (GCF) from each pair. This step reduces the complexity of the expression even further.
greatest common factor (GCF)
The greatest common factor (GCF) is the largest factor that two or more numbers (or terms in a polynomial) have in common. Finding the GCF is essential because it allows us to simplify expressions and solve equations more effectively. For example, in the group For the second group After factoring out the GCF, which makes our job easier in the next steps.
polynomial expressions
Polynomial expressions are mathematical phrases involving a sum of powers in one or more variables multiplied by coefficients. They are one of the most common types of expressions in algebra. For instance, For example:
  • Constant: a number without any variable (e.g., 5)
  • Monomial: a term with a coefficient and a variable (e.g., 4b)
  • Binomial: two monomials added or subtracted (e.g., b^2 + 5b)
  • Trinomial: three monomials added or subtracted (e.g., b^2 + 5b - 4b)

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Most popular questions from this chapter

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