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Chapter 5: Problem 80

Factor by grouping. \(t s+5 t+2 s+10\)

Short Answer

Expert verified
(s + 5)(t + 2)

Step by step solution

01

- Group the terms

Group the terms in pairs to make it easier to factor. The given expression is:\[ts + 5t + 2s + 10\]Group the terms like this:\[(ts + 5t) + (2s + 10)\]
02

- Factor out the GCF from each group

Factor out the greatest common factor (GCF) from each pair of terms. For the first group, \(ts + 5t\), the GCF is \(t\). For the second group, \(2s + 10\), the GCF is \(2\):\[t(s + 5) + 2(s + 5)\]
03

- Factor out the common binomial

The expressions have a common binomial factor, \(s + 5\). Factor this out to get the final factored form:\[(s + 5)(t + 2)\]

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Key Concepts

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

greatest common factor
To grasp factoring by grouping, you first need to understand the concept of the greatest common factor (GCF). The GCF of two numbers or terms is the largest number or term that can divide both without leaving a remainder. Identifying the GCF can simplify expressions and help in factoring. For example, in the term group \(ts + 5t\), both terms share a common factor: \(t\). Factoring out \(t\) yields \(t(s + 5)\). Similarly, for the group \(2s + 10\), the GCF is \(2\), giving us \(2(s + 5)\). Recognizing and factoring out the GCF is a crucial step in solving these types of problems.
binomial factor
A binomial factor is an expression consisting of two terms. It plays a key role in factoring by grouping. In our example, \(ts + 5t + 2s + 10\), after grouping and factoring out the GCFs, both groups contained a common binomial factor: \(s + 5\). This commonality allows us to factor the entire expression further. The expression goes from \(t(s + 5) + 2(s + 5)\) to \((s + 5)(t + 2)\). This step is pivotal in simplifying expressions and solving polynomial equations.
factoring
Factoring involves breaking down an expression into simpler components (factors) that, when multiplied together, give the original expression. It's essential in solving equations and simplifying expressions. Our problem \(ts + 5t + 2s + 10\) uses the method of factoring by grouping:
  • First, group terms to make factoring easier: \((ts + 5t) + (2s + 10)\)
  • Then, factor out the GCF from each group: \(t(s + 5)\) and \(2(s + 5)\)
  • Finally, factor out the common binomial term: \((s + 5)(t + 2)\)
This method is practical for polynomial expressions with four or more terms.

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