Problem 22 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$30 b^{3}-5 b$$

Step-by-Step Solution

Answer

The short answer is: \(5b(6b^2 - 1)\).

Step 1: Identify the terms in the expression

First, we need to identify the two terms in the expression, which are: 1. \(30 b^3\) 2. \(-5 b\)

Step 2: Determine the GCF of the coefficients and variables separately

We can break both terms into their numerical coefficients and variables: 1. \(30 b^3 = 3 \cdot 2 \cdot 5 \cdot b^3\) 2. \(-5 b = -5 \cdot b\) The GCF of the coefficients 30 and -5 is 5, and the GCF of the variables \(b^3\) and b is b. Therefore, the GCF of the terms is \(5b\).

Step 3: Factor out the GCF

Now, we can factor out the GCF, \(5b\), from both terms in the expression. We divide both terms by the GCF and multiply the GCF by the resulting expression: \(30 b^3 - 5b = 5b(6b^2 - 1)\)

Step 4: Check the answer

To check our answer, we can distribute the GCF back into the expression to see if we get the original expression: \(5b(6b^2 - 1) = 30 b^3 - 5 b\) Since the expression matches the original expression, we have factored correctly. So, the final factored expression is: \(5b(6b^2 - 1)\)

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the pieces that make up a whole. In algebra, a polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Factoring is the process of breaking down these polynomials into simpler, multiplied expressions. One important tool in factoring polynomials is the greatest common factor (GCF).Factoring out the GCF means identifying and isolating the largest factor that both terms share. By dividing each term by the GCF, we simplify the polynomial, making it easier to work with. For example, in the expression \(30b^3 - 5b\), we factor out the GCF \(5b\) to simplify the expression into \(5b(6b^2 - 1)\). This process makes solving and simplifying future operations with that polynomial more manageable.

Coefficients and Variables

In algebra, terms within expressions are made up of coefficients and variables. Coefficients are the numerical part of a term, while variables are the letters that represent unknown or changeable numbers. Understanding the role of both helps us manipulate and simplify expressions efficiently.

• Coefficients: These are the numbers in front of the variables. In our example, 30 and -5 are coefficients of the terms in the expression.
• Variables: In this context, variables represent unknowns (like \(b\) in our expression). They can have powers, such as \(b^3\), which indicates that the variable is raised to the third power.
• Combining Coefficients and Variables: When finding the GCF, we can break down the coefficients into their prime factors and similarly consider the lowest power of the variables.

Knowing how to separate and recombine coefficients and variables is crucial for factoring and simplifying algebraic expressions.

Algebraic Expressions

Algebraic expressions consist of terms that can include constants, variables, and the operations of addition, subtraction, multiplication, and division. Understanding how to navigate these expressions is essential for solving algebra problems.Each term in an expression can be separated by addition or subtraction and is composed of a coefficient and a variable. For example, \(30b^3 - 5b\) consists of two terms. To work with these types of expressions:

• Identify individual terms: In \(30b^3 - 5b\), we have two terms—each made up of a coefficient and a variable component.
• Apply operations: Operations such as factoring involve finding common factors or rearranging terms to simplify the expression.
• Simplification: By reordering and manipulating terms, expressions become easier to manage, solve, or further analyze.

Recognizing and understanding the structure of algebraic expressions helps to confidently apply mathematical techniques like factoring to solve complex equations and problems.