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Chapter 6: Problem 2

Factor completely. If the polynomial is not factorable, write prime. $$ a^{2}+5 a+a b $$

Short Answer

Expert verified
The expression factored completely is \(a(a + b + 5)\).

Step by step solution

01

Identify Common Terms

Look at each term in the polynomial \(a^2 + 5a + ab\). We notice that \(a\) is a common factor in all three terms.
02

Factor Out the Common Term

Factor \(a\) out of each term. This gives us \(a(a + 5 + b)\).
03

Rearrange the Terms

Reorder the expression inside the parentheses to make it clearer: \(a(a + b + 5)\). This makes it easier to check for further factoring.
04

Check for Complete Factorization

Inspect \(a + b + 5\) to determine if it can be further factored. Since \(a + b + 5\) consists of terms that cannot be combined or factored further, this is the complete factorization.

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Key Concepts

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

Common Factor
When tackling polynomial expressions, the first step in factoring is to spot the common factor among the terms. This essentially means looking for a term that appears in each part of the polynomial. In the given expression \[a^{2} + 5a + ab\], if you closely observe each term鈥� \(a^2\), \(5a\), and \(ab\)鈥攜ou will see that the variable \(a\) is present in all three. Identifying this common factor is crucial because it simplifies the expression and sets the foundation for further factorization.Finding a common factor involves:
  • Looking at the coefficients and variables in each term,
  • Making note of any numbers or variables that are shared among terms.
By simplifying through factoring out common terms, you make complex expressions more manageable.
Expression Rearrangement
Once you factor out the common factor in a polynomial, the next handy step is expression rearrangement. This means organizing the resulting expression to make it clearer or simpler to work with. In our expression, after factoring out \(a\), we had: \[a(a + 5 + b)\]. By rearranging the terms inside the parentheses, you get \[a(a + b + 5)\]. Rearranging an expression involves organizing terms in a visible and structured order that often makes the expression easier to interpret or check further for factorization. It can help:
  • Highlight any patterns or additional common factors,
  • Facilitate the recognition of identities or simplifications.
Just remember, rearrangement is all about clarity and organization鈥攊t doesn鈥檛 alter the values or relationships in the expression.
Complete Factorization
After factoring out a common term and rearranging the expression, you aim to achieve a completely factorized form. This is when the expression is reduced to a point where no further even division occurs among its elements. The polynomial \[a(a + b + 5)\] is indicating its complete factorization because the remaining expression \(a + b + 5\) contains terms that neither share additional common factors nor fit any straightforward factorization patterns.Complete factorization is your goal because:
  • It represents a final, simplified state of the polynomial,
  • It ensures no possible further division or simplification can be applied,
  • It helps uncover the polynomial's roots or solutions, should they be needed for solving equations.
In practice, this means evaluating each part of the expression critically, ensuring all potential factor paths have been followed, ensuring you have indeed reached a "prime" state.

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