Problem 67 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 2 y-8+x y-4 x $$

Step-by-Step Solution

Answer

The polynomial is factored as \((2 + x)(y - 4)\).

Step 1: Rearrange and Group Terms

First, rearrange the terms to organize them for grouping. The polynomial \(2y - 8 + xy - 4x\) can be rearranged as \((2y + xy) + (-8 - 4x)\). Group the terms as shown for easier factoring.

Step 2: Factor Common Factors in Each Group

For the first group \((2y + xy)\), factor out \(y\) to get \(y(2 + x)\). For the second group \((-8 - 4x)\), factor out \(-4\) to get \(-4(2 + x)\). Now the expression looks like \(y(2 + x) - 4(2 + x)\).

Step 3: Factor Out the Common Binomial

Notice that both terms contain the common binomial \((2 + x)\). Factor \((2 + x)\) out of the expression: \((2 + x)(y - 4)\). This is the fully factored form of the polynomial.

Key Concepts

Polynomial Grouping

Polynomial grouping is a useful technique when dealing with polynomials that have more than three terms. This method involves rearranging the polynomial into groups that can then be factored separately. By focusing on smaller sub-expressions, polynomial grouping allows for more sophisticated factorization.

In the context of a four-term polynomial, such as the one given in the exercise, the goal is to strategically arrange and group terms so that each group shares a common factor.

• Start by observing terms that naturally pair together, often based on coefficients or variable parts.
• For example, in the polynomial transformed from \(2y - 8 + xy - 4x\) to \((2y + xy) + (-8 - 4x)\), terms are grouped to facilitate further factorization.

By breaking down a complex expression into grouped parts, you can simplify it into a product of simpler factors.

Common Factors

Recognizing common factors is a key step in the polynomial grouping method. A common factor is a term or number that is multiplied by every term within a polynomial group. Identifying these makes the factorization process much smoother.

When assessing each group, look for variables, constants, or coefficients shared across all terms.

• In the example \((2y + xy) + (-8 - 4x)\), the common factor in the first group \(2y + xy\) is \(y\), giving us \(y(2 + x)\).
• The common factor in the second group \(-8 - 4x\) is \(-4\), yielding \(-4(2 + x)\).

Factoring out these common elements reduces the original expression into simpler, more manageable parts.

Binomial Factor

A binomial factor comes into play when, after grouping and factoring by common factors, you notice that the resulting groups share another factor – usually a binomial. This shared factor can be factored again, leading to the simplification of the expression.

In working with the expression \( y(2 + x) - 4(2 + x) \), note that the term \((2 + x)\) appears in both expressions. This repetition means \((2 + x)\) is a binomial factor.

• We can factor the binomial \((2 + x)\) out of the expression, linking the grouped terms into a single line: \((2 + x)(y - 4)\).
• This step drastically simplifies the polynomial, showing the power of binomial factoring.

Employing binomial factorization in this manner results in a cleaner solution and often reveals insights about the polynomial's structure.