Problem 37 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$m(n-12)+8(n-12)$$

Step-by-Step Solution

Answer

The GCF factored expression for the given expression \(m(n-12)+8(n-12)\) is \((n-12)(m+8)\).

Step 1: Identify the terms and their factors

First, we need to identify the terms in the given expression and find their factors: $$m(n-12)+8(n-12)$$ The terms are: 1. \(m(n-12)\) 2. \(8(n-12)\) Factors of the term \(m(n-12)\) are: \(m\) and \((n-12)\) Factors of the term \(8(n-12)\) are: \(8\) and \((n-12)\)

Step 2: Identify the greatest common factor (GCF)

Now, we need to find the greatest common factor between the terms. We can identify that both terms have the \((n-12)\) factor in common. GCF: \((n-12)\)

Step 3: Factor out the GCF

Next, we will factor out the GCF from the given expression by dividing each term by the GCF found in the previous step. $$m(n-12)+8(n-12) \implies (n-12)(\dfrac{m(n-12)}{n-12} + \dfrac{8(n-12)}{n-12})$$ This simplifies to: \((n-12)(m+8)\)

Step 4: Check the answer

To check our answer, we can distribute the GCF back to the terms inside the parenthesis and it should equal the original expression: $$(n-12)(m+8) \implies n(m+8)-12(m+8)$$ Simplify: $$nm+8n-12m-96$$ Rearrange: $$m(n-12)+8(n-12)$$ As it matches the original expression, our factored form is correct: GCF Factored Expression: \((n-12)(m+8)\)

Key Concepts

Factoring Expressions

Factoring expressions is like breaking down a big problem into smaller pieces to understand it better. When you look at an expression like \(m(n-12) + 8(n-12)\), you can see that we want to find a common part that both terms can share. This common part is called the "greatest common factor" (GCF). By factoring, you're simplifying the expression, making it easier to work with. Imagine you have a big box, and inside, there's another box that fits into two different sections perfectly. The box "(n-12)" is like that inner box. By pulling it out, you make the other sections, "m" and "8," clearer and separate, as shown when we rewrite the expression as: - \((n-12)(m+8)\) This way, you're expressing the problem compactly, but with visible parts. Factoring expressions like this not only helps in solving problems but also in understanding the structure of algebraic relationships.

Algebraic Expressions

Algebraic expressions are like puzzles made up of numbers, letters, and operations. Each part of an algebraic expression, called a term, helps form the overall "puzzle". These expressions help us neatly express problems so they can be solved step-by-step.Take the original expression \(m(n-12) + 8(n-12)\) from our exercise. Here, it consists of two terms, \(m(n-12)\) and \(8(n-12)\).

• The first term has the variable "m" next to the expression "(n-12)".
• The second term has the number "8" next to the same expression "(n-12)".

By spotting these patterns, you can find ways to manipulate them to make solving problems easier. Algebraic expressions are a way to tidy up and rethink complicated mathematical problems by breaking them down into their component parts.

Polynomial Factorization

Polynomial factorization is the method of breaking a polynomial into simpler parts, called "factors", that when multiplied together give back the original polynomial. Think of this as trying to find smaller building blocks that, together, make up the whole structure. When you have the expression \(m(n-12) + 8(n-12)\), the common factor \((n-12)\) is one such building block.The factorization steps are:

• Identify the GCF, which can be seen in both terms, resulting in \((n-12)\).
• Divide each term by this GCF, so the remaining factors are simplified to \((m+8)\).
• Put these pieces together: \((n-12)(m+8)\).

This method allows you to simplify polynomials just like breaking down numbers into their prime factors simplifies arithmetic. By doing this, you make algebraic problems easier to solve and understand. It's like uncovering a neat and tidy form of what may seem a complicated mess at first glance.