Problem 44 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(6 m(m-5)-7(m-5)\)

Step-by-Step Solution

Answer

(m-5)(6m-7)

Step 1: Identify the common factor

Look at both terms in the polynomial: The terms are: 1. 6m(m-5) 2. -7(m-5) Notice that (m-5) is common in both terms.

Step 2: Factor out the common factor

Factor the common term (m-5) outside the polynomial. This turns the expression into:(m-5)[6m - 7]

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is the largest factor that divides two or more numbers or terms. When factoring polynomials, finding the GCF can simplify expressions and make them easier to work with.
To identify the GCF in a polynomial, examine each term and determine the largest factor common to all terms.
In our example, we have the polynomial:

• 6m(m-5)
• -7(m-5)

Notice that the expression (m-5) appears in both terms. Therefore, (m-5) is our GCF. Once we identify the GCF, we can factor it out from the polynomial, simplifying our expression.

Factoring Out

Factoring out involves taking the GCF and extracting it from each term in the polynomial. This process simplifies the expression and reveals a simpler form.
Let's apply this to our example:
The given polynomial is:

• 6m(m-5) - 7(m-5)

First, we identified the (m-5) as our GCF.
Now, we factor out (m-5) from each term: We have: (m-5)[6m - 7]
By factoring out, we group the common factor outside the expression, leaving us with a simplified polynomial.
This process is crucial in simplifying algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operators that represent a value or relationship.
Understanding how to manipulate these expressions is fundamental for solving equations and simplifying terms.
In our polynomial example, we are given:

• 6m(m-5) - 7(m-5)

By factoring out, we simplify the expression into a cleaner, easier-to-understand form:

• (m-5)[6m - 7]

This new expression shows the relationship between the factors more clearly.
Recognizing patterns and common factors in algebraic expressions allows us to perform transformations, solve equations, and understand mathematical relationships deeply.
Algebraic expressions are the building blocks of algebra, and mastering their manipulation is key to success in mathematics.