Problem 71 - Elementary Algebra

Question

Factor. Either factor out the greatest common factor, factor by grouping, use the guess and check method, or use the \(a c\) method. $$ 6 m p+3 p w-8 m-4 w $$

Step-by-Step Solution

Answer

(2m + w)(3p - 4)

Step 1: Group the Terms

Group the expression in a way that allows for factoring by grouping: ewline (6mp + 3pw) - (8m + 4w)

Step 2: Factor Out the Common Factors

Factor out the greatest common factor from each group: ewline 3p(2m + w) - 4(2m + w)

Step 3: Factor by Grouping

Notice that each group contains a common factor ewline (2m + w). Factor this out: ewline (2m + w)(3p - 4)

Key Concepts

Greatest Common Factor

Sometimes you can simplify an algebraic expression by identifying the Greatest Common Factor (GCF) of its terms. The GCF is the largest factor shared by all terms in the expression. Finding the GCF involves looking for the highest number and variables that divide each term.
For example, in the expression \(6mp + 3pw - 8m - 4w\), we can group the terms to make this easier:

• Group 1: \(6mp + 3pw\)
• Group 2: \(8m + 4w\)

In each group, you'll notice that each pair of terms shares a factor that can be factored out:

• For \(6mp + 3pw\), the GCF is \(3p\).
• For \(8m + 4w\), the GCF is \(4\). The minus sign changes the GCF to \(-4\).

After factoring out these common factors, the expression looks cleaner, and further factoring steps become more accessible.

Factoring by Grouping

Factoring by grouping is a handy technique for simplifying complex algebraic expressions. This method works well when dealing with four or more terms. Grouping terms allows you to identify common factors more easily.
Let's take the expression mentioned earlier: \(6mp + 3pw - 8m - 4w\). Our first step is to group the terms:

• Group 1: \(6mp + 3pw\)
• Group 2: \(8m + 4w\)

Next, factor out the GCF from each group:

• For \(6mp + 3pw\), the GCF is \(3p\), so we factor out 3p to get \(3p(2m + w)\).
• For \(8m + 4w\), the GCF is \(4\), making it \(-4(2m + w)\).

After factoring each pair, we notice both groups share a common factor again: \((2m + w)\). We factor this out too:
\((2m + w)(3p - 4)\).

Algebraic Methods

Several algebraic methods exist to factor expressions, making algebra easier to understand. Choosing the right method for your problem improves your efficiency and understanding.
You've encountered a few techniques already, like factoring the Greatest Common Factor (GCF) and grouping. Let's review these and some other tools:

• Greatest Common Factor: Find the largest factor common to all terms and factor it out.
• Factoring by Grouping: Split the expression into groups and factor out the GCF from each group.
• AC Method: For quadratic expressions of the form \(ax^2 + bx + c\), this method helps break down the middle term for easier factoring.
• Difference of Squares: Recognize when an expression fits the pattern \(a^2 - b^2\) and use the formula \((a + b)(a - b)\).

Mastery of these algebraic methods significantly boosts your equation-solving skills and confidence.