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Chapter 6: Problem 32

Factor out the greatest common monomial factor from the polynomial. $$ 32 a^{5}-2 a^{3}+6 a $$

Short Answer

Expert verified
The simplified form of the polynomial by factoring out the greatest common monomial factor is \(2a(16 a^{4} - a^{2} + 3)\).

Step by step solution

01

Identify the GCF

The greatest common factor (GCF) of the coefficients 32, -2, and 6 is 2. In other words, 2 is the largest number that can divide each of these without a remainder.
02

Identify the lowest power of 'a'

The variable 'a' appears in each term with varying powers. To identify the lowest common power, find the smallest exponent from \(a^{5}\), \(a^{3}\), and \(a^{1}\) (assuming \(a\) as \(a^{1}\)). In this case, the lowest power of 'a' is 1.
03

Factor out the GCF and the lowest power of 'a'

Now, factor out the GCF and the lowest power of 'a' from each term of the polynomial. The polynomial then becomes: \(2a(16 a^{4} - a^{2} + 3)\). Each term within the parentheses is the result of dividing the original terms by \(2a\), the common factor.

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Key Concepts

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

Understanding the Greatest Common Factor
The Greatest Common Factor, or GCF, is a crucial concept in factoring polynomials. It represents the largest number that can divide each term in the polynomial without leaving a remainder. For efficient factoring, identifying the GCF helps to simplify expressions and make further operations easier. Let's break it down:
  • The GCF applies not only to coefficients, which are the numerical parts of the terms, but also to variables when they appear in every term.
  • To find the GCF of coefficients, list the factors of each number and choose the largest factor that appears in all lists.
  • When variables are involved, determine the smallest power of the variable present in all terms to be part of the GCF.
In our example, the coefficients 32, -2, and 6 all share 2 as their GCF. This is the largest number that divides each of them evenly.
Decoding Monomial Factors
Monomial factors are single terms consisting of a coefficient multiplied by variables raised to powers. Identifying monomial factors in a polynomial helps to simplify or reduce expressions, making them easier to work with. Here's how you can think about it:
  • Each monomial factor has a singular element and involves multiplication alone: for example, 2a.
  • To factor out a monomial, find the common coefficient and the lowest power of every variable that appears across all terms in the original expression.
  • Upon factoring out the monomial, each term in the polynomial should become a simpler form where calculations are straightforward.
In the original exercise, 2a is the monomial that can be factored out, simplifying each term into a more manageable expression.
Working with Algebraic Expressions
Algebraic expressions are mathematical phrases that can involve numbers, letters (variables), and operations. Learning how to manage and simplify these expressions is a fundamental skill in algebra. Consider the following points when working with algebraic expressions:
  • Unlike simple arithmetic, algebraic expressions can represent a wide range of values depending on the variable input.
  • You can add, subtract, multiply, and divide these expressions, as well as factor them to unravel complex terms.
  • Factoring is a common way to reduce expressions to simpler forms or solve equations, focusing on extracting any common elements first.
In the provided polynomial example, factoring out the greatest common monomial factor helps transform the expression into a form that highlights residual terms in relation to the factorized ones.

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