Problem 27 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 30 x-15 $$

Step-by-Step Solution

Answer

The GCF is 15; factored form: \(15(2x - 1)\).

Step 1: Identify the GCF

To factor out the Greatest Common Factor (GCF), we first identify the common factor for the coefficients of the terms in the polynomial. Here, the terms are \(30x\) and \(-15\). The coefficients are \(30\) and \(-15\). The GCF of \(30\) and \(15\) is \(15\).

Step 2: Factor out the GCF

Once we have identified the GCF, which is \(15\), we can factor it out from each term. This means we write each term as a product of the GCF and another factor. \[30x - 15 = 15(2x) - 15(1)\]

Step 3: Express the Polynomial with the GCF Factored Out

Combine the terms inside the parentheses to express the polynomial with the GCF factored out completely. \[15(2x - 1)\] represents the polynomial with the GCF factored out.

Key Concepts

Understanding Polynomials

A polynomial is a mathematical expression that involves a sum of powers in one or more variables multiplied by coefficients. In simple terms, it's like a phrase made of numbers, variables like \(x\), \(y\), etc., and operations such as addition, subtraction, and multiplication.
For example, in our original exercise, \(30x - 15\) is a polynomial. The polynomial consists of two terms, "\(30x\)" and "\(-15\)".
Each term is composed of coefficients and variables:

• "30" is the coefficient of "\(x\)".
• "-15" is a constant term since it doesn’t have a variable associated with it.

This concept will help you see how polynomials can be simplified or broken down using techniques such as factoring. Polynomials form the foundation of many algebraic operations and are something you'll encounter frequently in algebra.

Factoring and the Greatest Common Factor

Factoring in mathematics is the process of breaking down a complex expression into a product of simpler factors. One common technique is factoring out the Greatest Common Factor (GCF), which helps simplify polynomials by extracting the largest common factor from each of the terms.
In our exercise, we had the polynomial \(30x - 15\). To factor it, we first needed to find the GCF of the coefficients "30" and "15", which is "15".

• You divide each term by the GCF.
• Write the polynomial as a product of the GCF and the simplified terms.

This gives us: \[15(2x - 1)\].
By finding the GCF, we simplify the polynomial, making it easier to work with in further mathematical operations like adding, subtracting, or even differentiating polynomials.

Applying Algebra in Simplifying Polynomials

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. These symbols, usually letters, represent values or numbers in equations and expressions.
Simplifying expressions, such as polynomials, is a key part of algebra, and factoring is one of the tools we use in this process.
When we factor out the GCF from a polynomial, we are applying algebraic principles to rewrite it in a more simplified form, as seen with \(30x - 15\) being reduced to \[15(2x - 1)\].

• This makes complex expressions more manageable.
• Factoring helps in solving algebraic equations where setting expressions to zero is needed.

Understanding how to factor polynomials is a valuable skill in algebra—especially because it aids in problem-solving and equation manipulation, making algebraic problems much more approachable.