Problem 54 - 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 as \((x - 4y)(x + y)\).

Step 1: Identify the Greatest Common Factor (GCF)

Examine each term in the trinomial: \(x^2\), \(-3xy\), and \(-4y^2\). The terms \(x^2\) and \(-4y^2\) do not have a common factor with \(-3xy\), thus there is no GCF other than 1. Proceed to factor the trinomial as is.

Step 2: Rewrite Trinomial in a Factorable Form

We have \(x^2 - 3xy - 4y^2\) as a simple quadratic in terms of \(x\) and \(y\). It is already in the standard form \(ax^2 + bxy + cy^2\) with \(a = 1\), \(b = -3\), and \(c = -4\). Now let's find two numbers that multiply to \(ac = -4\) and add to \(b = -3\).

Step 3: Find Numbers to Factor the Middle Term

Solve for two numbers whose product is \(-4\) and sum is \(-3\). The numbers are \(-4\) and \(1\), since \(-4 \times 1 = -4\) and \(-4 + 1 = -3\). Use these numbers to split the middle term.

Step 4: Split Middle Term and Group Terms

Rewrite \(-3xy\) as \(-4xy + 1xy\) to get two pairs: \(x^2 - 4xy + xy - 4y^2\). Now, group the terms: \((x^2 - 4xy) + (xy - 4y^2)\).

Step 5: Factor by Grouping

Look at each group: From \(x^2 - 4xy\), factor out \(x\) to get \(x(x - 4y)\). From \(xy - 4y^2\), factor out \(y\) to get \(y(x - 4y)\). This gives the expression: \( x(x - 4y) + y(x - 4y)\).

Step 6: Factor Out Common Binomial Factor

Notice \((x - 4y)\) is common in both groups. Factor \((x - 4y)\) out to get \((x - 4y)(x + y)\). Thus, the trinomial is completely factored.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key concept when factoring trinomials, although not every trinomial will have a GCF other than 1. In this case, the exercise is teaching us that before jumping into complex factoring methods, we should always first check if there is a number or variable common to all terms. The GCF is the largest factor that divides each of the terms. It simplifies the expression first, making further factoring easier.

• In the trinomial given, each term was checked for common factors: \(x^2\), \(-3xy\), and \(-4y^2\).
• If a GCF had existed here (like a variable common to all terms), it would have been factored out first.
• However, in this exercise, there was no common factor besides 1, so the trinomial was left as is for subsequent steps.

Understanding the role of the GCF helps in simplifying expressions early on, which often leads to quicker and more accurate answers. Always look for GCF as your first step to factor any polynomial.

Factoring by Grouping

Factoring by grouping is a valuable strategy for breaking down polynomials into simpler, easily manageable pieces. This method involves dividing the expression into groups that can be factored separately, and then factoring out the common factors from these groups.
When dealing with our example trinomial, breaking it into groups helps find a common binomial factor.

• We start with \(x^2 - 3xy - 4y^2\). The middle term \(-3xy\) was split using numbers \(-4\) and \(1\) because they multiply to \(-4\) (product of first and last coefficients) and add to \(-3\).
• The trinomial was restructured as \(x^2 - 4xy + xy - 4y^2\).
• This divided the terms into two pairs: \((x^2 - 4xy)\) and \((xy - 4y^2)\).

In each group, factors were extracted: the first group yielded \(x(x-4y)\), and the second \(y(x-4y)\), illustrating how grouping and factoring simplifies complex expressions.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, usually in the form \(ax^2 + bx + c\). Here, we face a slightly different form \(ax^2 + bxy + cy^2\), which still follows the essential characteristics of a quadratic.
Let’s explore the structure of this quadratic expression within the trinomial factoring context.

• The given quadratic, \(x^2 - 3xy - 4y^2\), combines both \(x\) and \(y\) as variables, which influences how we think about factoring.
• Identifying terms as \(a = 1\), \(b = -3\), \(c = -4\) reflects the quadratic form we are accustomed to, factoring it follows the same principle.
• Finding factors that multiply to the product \(a \cdot c = -4\) and add up to \(b = -3\) allows us to approach factoring similarly to single-variable quadratics.

Thus, dealing with quadratic expressions with mixed variables requires attention to how each term contributes to the expression's overall structure. This leads to successful factoring, even in what may initially seem complex.