Problem 41 - Elementary and Intermediate Algebra

Question

Factor. See Example 6 or Example 11. $$x^{2}-6 x y-7 y^{2}$$

Step-by-Step Solution

Answer

The factorization is \((x - 7y)(x + y)\).

Step 1: Understand the Problem

We need to factor the quadratic expression \(x^2 - 6xy - 7y^2\). This is a quadratic in terms of \(x\) and \(y\). To factor it, we need to find two binomials that multiply together to give the original expression.

Step 2: Identify Quadratic Form

Recognize this expression as a quadratic trinomial in the form \(ax^2 + bxy + cy^2\) where \(a = 1\), \(b = -6\), and \(c = -7\). Our goal is to split the middle term in such a way we can factor by grouping.

Step 3: Find Two Numbers

We look for two numbers that multiply to \(a imes c = 1 imes -7 = -7\) and add up to \(b = -6\). These numbers are \(-7\) and \(1\) because \(-7 + 1 = -6\) and \(-7 imes 1 = -7\).

Step 4: Rewrite the Expression

Rewrite \(-6xy\) as \(-7xy + xy\). The expression becomes \(x^2 - 7xy + xy - 7y^2\).

Step 5: Group the Terms

Group the terms into two binomials: \((x^2 - 7xy)\) and \((xy - 7y^2)\).

Step 6: Factor Each Group

Factor out the greatest common factor from each group: from \(x^2 - 7xy\), factor out \(x\), getting \(x(x - 7y)\). From \(xy - 7y^2\), factor out \(y\), getting \(y(x - 7y)\).

Step 7: Final Factorization

Since both groups contain the common binomial \((x - 7y)\), factor it out: \((x - 7y)(x + y)\). So, the factorized form of the expression is \((x - 7y)(x + y)\).

Key Concepts

Quadratic Trinomial

A quadratic trinomial is a polynomial expression that consists of three terms with a degree of two. The general form is given by \(ax^2 + bxy + cy^2\). In this expression, the terms involve two variables: \(x\) and \(y\). Understanding and recognizing the structure of a quadratic trinomial is crucial because it sets the stage for various factoring techniques. In the exercise, we have the quadratic expression \(x^2 - 6xy - 7y^2\) where \(a = 1\), \(b = -6\), and \(c = -7\).

• The term \(x^2\) represents the quadratic term for \(x\).
• The middle term \(-6xy\) involves both variables, acting as a bridge between the two single-variable terms.
• The term \(-7y^2\) is the quadratic term for \(y\).

Recognizing these parts helps in utilizing appropriate methods to manipulate and simplify the expression further.

Binomial Factorization

Binomial factorization involves expressing a quadratic trinomial as the product of two simpler binomial terms. The ultimate goal with binomial factorization is to break down complex expressions into manageable pairs of expressions that multiply to yield the original trinomial. In the provided exercise, the expression \(x^2 - 6xy - 7y^2\) must be expressed as the product \((x - 7y)(x + y)\). Here's how we achieve this:

• First, rewrite the trinomial so that we can separate terms and look for common factors. Here, we split the middle term \(-6xy\) into \(-7xy + xy\).


• Next, group the terms into pairs: \((x^2 - 7xy)\) and \((xy - 7y^2)\).


• Find the greatest common factor within each group. Factor \(x\) from the first group resulting in \(x(x - 7y)\) and \(y\) from the second group to get \(y(x - 7y)\).


• Finally, because both groups share a common binomial \((x - 7y)\), you can factor that out, leaving us with the expression \((x - 7y)(x + y)\).

These steps simplify solving systems of equations involving quadratic trinomials by making them more approachable through breaking down.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by two or more numbers or expressions. It is extremely useful when factoring expressions because it allows us to simplify complicated terms.In the context of factoring binomials and trinomials, identifying and factoring out the GCF is often a preliminary step. For the quadratic trinomial \(x^2 - 6xy - 7y^2\), determining the GCF involves concentrating on grouped terms:

• Consider the expression \(x^2 - 7xy\). The GCF in this case is \(x\), so we factor it out, leaving \(x(x - 7y)\).


• In the second group \(xy - 7y^2\), the GCF is \(y\). Factoring out \(y\), we obtain \(y(x - 7y)\).

Notice that, after factoring out the GCF in each group, both newly formed parts share a common factor: \((x - 7y)\). This enables further simplification through final factorization.
Using the GCF effectively simplifies expressions and primes them for easier manipulation in subsequent steps. Understanding and identifying the GCF is thus a vital skill in the factoring process.