Problem 28 - 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. $$ x^{2}-4 x y-77 y^{2} $$

Step-by-Step Solution

Answer

The trinomial factors completely to \((x + 7y)(x - 11y)\).

Step 1: Identify the GCF

First, look for a greatest common factor (GCF) in all the terms of the trinomial. In this case, each term does not share a common factor (other than 1), so we do not need to factor out anything at this step.

Step 2: Determine the Form

Identify if the trinomial is in the form of a quadratic equation, which here is in standard form: \(ax^2 + bxy + cy^2\). For the given trinomial \(x^2 - 4xy - 77y^2\), we have \(a = 1\), \(b = -4\), and \(c = -77\).

Step 3: Look for Factor Pairs

We need factors of \(a imes c\) (which is \(1 imes (-77) = -77\)) that add up to \(b\) (which is \(-4\)). The factors of \(-77\) that will sum to \(-4\) are \(7\) and \(-11\).

Step 4: Break Down the Middle Term

Rewrite the trinomial by splitting the middle term using the factors found: \(x^2 + 7xy - 11xy - 77y^2\).

Step 5: Grouping

Group the terms to factor by grouping: \((x^2 + 7xy) + (-11xy - 77y^2)\).

Step 6: Factor by Grouping

Factor each group independently: from \((x^2 + 7xy)\), factor out \(x\) to get \(x(x + 7y)\) and from \((-11xy - 77y^2)\), factor out \(-11y\) to get \(-11y(x + 7y)\).

Step 7: Combine the Factors

Since both groups contain \((x + 7y)\), factor \((x + 7y)\) out: \((x + 7y)(x - 11y)\). Thus, the trinomial \(x^2 - 4xy - 77y^2\) is completely factored as \((x + 7y)(x - 11y)\).

Key Concepts

greatest common factor (GCF)

In mathematics, the greatest common factor (GCF) of two or more numbers is the largest positive integer that divides each of the numbers without a remainder. It's essentially the biggest "piece" that all the numbers share evenly. When dealing with polynomials, like trinomials, identifying the GCF involves finding the largest polynomial that divides each term of the expression.

• If every term of a trinomial has a common factor, it's crucial to factor it out first as it simplifies the polynomial and makes further factoring easier.
• In some cases, like the trinomial $x^2 - 4xy - 77y^2$, terms don't share a common factor other than 1, so we proceed without factoring anything out initially.

Recognizing the GCF is a valuable step because it reduces the complexity of an equation, streamlining the process of solving or factoring it down the line. Factoring the GCF is often the first step in problems involving factoring polynomials.

quadratic equations

Quadratic equations are fundamental in algebra, represented in the standard form: \( ax^2 + bx + c \). They form a parabola when graphed and can often be solved by factoring, among other methods like completing the square or using the quadratic formula.
In the case of a trinomial like \( x^2 - 4xy - 77y^2 \), it's expressed as a quadratic in terms of \(x\) and \(y\). Here:

• The coefficient \(a = 1\)
• The coefficient \(b = -4\)
• The coefficient \(c = -77\)

To solve or factor these equations, identify values for \( a \), \( b \), and \( c \) and look for factor pairs of \( ac \) (the product of \(a\) and \(c\)) that sum to \( b \). Understanding this basic structure is key for identifying on how a quadratic equation can be manipulated or simplified.

factor by grouping

Factor by grouping is a technique used in algebra to simplify polynomials by grouping terms that have common factors. This method is especially useful and effective when dealing with multi-term polynomials, like trinomials, that do not yield easily to other factoring methods.
Let’s take the trinomial \( x^2 - 4xy - 77y^2 \), which we want to factor. The steps to factor by grouping include:

• Rewrite the middle term, \(-4xy\), using two numbers that both multiply to \( ac = -77 \) and add to \(-4\). These numbers are \(7\) and \(-11\).
• The expression becomes \( x^2 + 7xy - 11xy - 77y^2 \).
• Group the terms: \( (x^2 + 7xy) + (-11xy - 77y^2) \).
• Factor out the greatest common factor from each group: \( x(x + 7y) \) and \(-11y(x + 7y) \).
• Notice both groups share a common binomial factor \(x + 7y\). Factor this out to get \( (x + 7y)(x - 11y) \).

This method allows complex expressions to be broken down into simpler binomial pairs, making solving, analyzing, or further simplifying the expression much more straightforward.