Problem 74 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 5 x y-15 x-6 y+18 $$

Step-by-Step Solution

Answer

The factored form is \((5x - 6)(y - 3)\).

Step 1: Group the Terms

Group the terms that can be factored separately. We can group the polynomial \(5xy - 15x\) and \(-6y + 18\).

Step 2: Find the Common Factor in Each Group

In the first group \(5xy - 15x\), the common factor is \(5x\). In the second group \(-6y + 18\), the common factor is \(-6\).

Step 3: Factor Out the Common Factors

Factor the common factor out of each group: \(5x(y - 3) - 6(y - 3)\).

Step 4: Factor Out the Common Binomial

Both terms now include the common binomial \((y - 3)\). Factor \((y - 3)\) out of the expression: \((5x - 6)(y - 3)\).

Step 5: Verify the Factors

Expand \((5x - 6)(y - 3)\) to check: \(5x(y-3) - 6(y-3) = 5xy - 15x - 6y + 18\), which is the original polynomial.

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials, especially useful when dealing with four-term polynomials.
This technique involves rearranging and grouping polynomial terms to make it easier to factor them.
Here's how it works:

• Step 1: Group Terms Wisely
  Identify pairs of terms in the polynomial that can be factored out.
  Choose groups that share a common factor. For instance, in the polynomial given in the exercise, you can group it as \(5xy - 15x\) and \(-6y + 18\).
• Step 2: Factoring within Groups
  Focus on each group separately and find the greatest common factor (GCF).
  In our case, \(5xy - 15x\) has a GCF of \(5x\), and \(-6y + 18\) has a GCF of \(-6\).

By effectively using factoring by grouping, you can simplify complex polynomials into neatly factored expressions, setting a foundation for solving equations.

Common Factors

Common factors are central to the process of factoring by grouping.
They are elements that are shared across multiple terms in a polynomial.
When factoring a polynomial by grouping, you look for the greatest common factor that can be pulled out from each individual group.
This involves:

• Finding the GCF: Examine each group of terms separately. Identify and extract the largest factor common to all terms. In \(5xy - 15x\), the common factor is \(5x\).
• Applying the GCF: Once you identify the GCF, you factor it out from the group. As demonstrated, \(5xy - 15x\) becomes \(5x(y - 3)\) after factoring out \(5x\).

Finding common factors simplifies the polynomial, making it easier to manage and solve. Recognizing these factors allows for more efficient algebraic manipulation.

Four-Term Polynomials

Four-term polynomials can often be intimidating, but they present unique opportunities for factoring, primarily by grouping.
This type of polynomial is characterized by having four distinct terms, such as \(5xy - 15x - 6y + 18\).
For these polynomials, factoring by grouping is particularly effective because:

• Grouping Pairs of Terms: The four terms are divided into two pairs. By choosing pairs with common factors—like \(5xy - 15x\) and \(-6y + 18\)—the polynomial becomes easier to work with.
• Facilitating Further Factoring: Once grouped, you can factor out the GCF from each pair, simplifying the expression significantly.

By focusing on the structure and symmetry of four-term polynomials, you can apply systematic factoring techniques, effectively solving challenging algebraic problems.