Problem 76 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 6 m^{2}-5 m n-6 m+5 n $$

Step-by-Step Solution

Answer

Factored as \((6m - 5n)(m - 1)\).

Step 1: Group the First Two Terms

Take the polynomial \(6m^2 - 5mn - 6m + 5n\) and group the first two terms. This gives us \((6m^2 - 5mn)\).

Step 2: Group the Last Two Terms

Now, group the last two terms of the polynomial, resulting in \((-6m + 5n)\).

Step 3: Factor Common Factors from Each Group

For the first group \((6m^2 - 5mn)\), factor out the common factor, which is \(m\): \(m(6m - 5n)\). For the second group \((-6m + 5n)\), factor out \(-1\) to get \(-1(6m - 5n)\).

Step 4: Identify Common Binomial Factor

Notice now that each group has \((6m - 5n)\) as a common factor: \(m(6m - 5n) - 1(6m - 5n)\).

Step 5: Factor Out the Common Binomial

The expression can now be written as \((6m - 5n)(m - 1)\) by factoring out the common binomial \((6m - 5n)\). This is the completely factored form.

Key Concepts

Factoring by Grouping

Factoring by grouping is a useful method to factor polynomials, especially those with four terms. The objective is to rearrange and group the terms in such a way that each group has a common factor, which can then be factored out. To apply this technique:

• Step 1: Identify and group terms. For a four-term polynomial, divide it into two pairs or groups of terms.
• Step 2: Factor each group separately by looking for common factors.
• Step 3: If both groups have a common binomial factor, factor this binomial out to simplify the expression.

Typically, if grouping doesn't reveal a common factor, then the polynomial might not be factorable by this method. This approach excels when a polynomial can be broken down into recognizable patterns or simpler products.

Binomial Factor

When factoring polynomials by grouping, the goal often is to identify a common binomial factor. A binomial is an algebraic expression with two terms, such as (6m - 5n) in our example. Here’s why identifying a binomial factor is important:

• It simplifies polynomial expressions to smaller components.
• All terms in the polynomial can sometimes be expressed in terms of this binomial factor.
• Rightly identifying a common binomial can easily resolve the polynomial into simpler factors.

Once you have a common factor like (6m - 5n), the next step is essential: factor it out. The remaining expression, (m - 1) for instance, reveals the entire structure of the polynomial. This process is key to simplification and further algebraic manipulation.

Common Factors in Polynomials

Common factors play a vital role in polynomial factoring. They are the numbers or terms that are common across multiple groups within the polynomial. Here's how to find and use them effectively:

• Identify: Look for terms that repeat across different parts of the polynomial expression.
• Factor Out: Pull out these common factors from individual groups to simplify the polynomial.
• Simplify: Once common factors are extracted, the polynomial breaks down into easier components.

Finding common factors requires a careful examination of the terms and often begins with inspecting coefficients and variables. In our example, noticing that m and -1 both link up with the binomial (6m - 5n) is crucial. Factoring them out results in a neater and more manageable expression.