Problem 27 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ x^{2}-3 x y-4 y^{2} $$

Step-by-Step Solution

Answer

The trinomial \( x^2 - 3xy - 4y^2 \) factors to \( (x + y)(x - 4y) \).

Step 1: Identify the Greatest Common Factor (GCF)

The expression given is \( x^{2} - 3xy - 4y^{2} \). Examine each term to determine if there is a common factor other than 1. The terms are \( x^2 \), \( -3xy \), and \( -4y^2 \); they do not have a common factor other than 1, so there is no GCF to factor out.

Step 2: Set Up for Factoring the Trinomial

Since the expression is a trinomial and there is no GCF except 1, rewrite it considering it in the form \( ax^2 + bxy + cy^2 \). Here, \( a = 1 \), \( b = -3 \), and \( c = -4 \). We need to factor this as a product of two binomials.

Step 3: Determine the Binomial Factors

To factor \( x^2 - 3xy - 4y^2 \), find two numbers whose product is \( ac = 1 \times -4 = -4 \) and sum is \( b = -3 \). These numbers are \(-4\) and \(1\).

Step 4: Decompose the Middle Term

Rewrite \(-3xy\) as \(-4xy + xy\). The expression becomes \( x^2 - 4xy + xy - 4y^2 \).

Step 5: Factor by Grouping

Group the terms: \( (x^2 - 4xy) + (xy - 4y^2) \). Factor each group: \( x(x - 4y) + y(x - 4y) \).

Step 6: Factor Out the Common Binomial

The common binomial factor is \( (x - 4y) \). Therefore, the expression can be factored as \( (x + y)(x - 4y) \).

Key Concepts

Greatest Common Factor

To start factoring any polynomial, we first look for the Greatest Common Factor (GCF). The GCF is the largest factor common to each term in the expression. In our trinomial, \(x^2 - 3xy - 4y^2\), we have three terms: \(x^2\), \(-3xy\), and \(-4y^2\). It's essential to check if any number or variable can evenly divide all these terms. For instance, examine the coefficients: 1, -3, and -4. There is no number other than 1 that can divide them all. Next, consider the variables: note that while \(x\) appears in the first two terms, it doesn't appear in the third. Similarly, \(y\) is absent in the first term. Thus, there is no common variable in all terms. Therefore, in this case, the GCF is simply 1, and we cannot factor anything out. This means we proceed to other factoring methods without simplifying the expression further.

Binomial Factors

Once it is determined that no GCF can be factored out other than 1, the next step is to express the trinomial as a product of two binomials. The standard form \(ax^2 + bxy + cy^2\) guides us, where in this instance, \(a = 1\), \(b = -3\), and \(c = -4\). The challenge is to find two numbers that both add to \(b\) and multiply to \(ac\). For \(x^2 - 3xy - 4y^2\), we calculate \(ac = 1 \times -4 = -4\), and we need the numbers whose product is \(-4\) and sum is \(-3\). We identify \(-4\) and 1 as these numbers:

• Sum: \(-4 + 1 = -3\)
• Product: \(-4 \times 1 = -4\)

Using these, the expression \(-3xy\) in our trinomial is rewritten as \(-4xy + xy\), setting up the equation for the grouping method.

Grouping Method

The grouping method is pivotal in breaking down a trinomial into a simpler factored form. We modify our trinomial into four terms for easier grouping. Using the numbers we obtained earlier, we rewrite \(-3xy\) as \(-4xy + xy\), changing our trinomial \(x^2 - 3xy - 4y^2\) into \(x^2 - 4xy + xy - 4y^2\).Grouping involves pairing terms to factor easily. Consider:

• Group 1: \(x^2 - 4xy\)
• Group 2: \(xy - 4y^2\)

For Group 1, factor out \(x\) to get \(x(x - 4y)\). For Group 2, factor out \(y\) to obtain \(y(x - 4y)\). Now the expression reads \(x(x - 4y) + y(x - 4y)\). Since both terms include \((x - 4y)\), factor it out as the common term, leading to the result \((x + y)(x - 4y)\). This showcases how grouping simplifies complexity, resulting in two clear binomial factors.