Problem 111 - College Algebra

Question

Factor completely. $$ (x-y)^{4}-4(x-y)^{2} $$

Step-by-Step Solution

Answer

The complete factored form of the provided term is \(-1(x - y)^2)(3(x - y)^2)\).

Step 1: Rewrite the Terms

Rewrite the equation so that ​\(4(x−y)^2\) is represented as ​\([2(x−y)^2]^2\). Then the equation becomes \((x-y)^4 - [2(x-y)^2]^2\).

Step 2: Apply Difference of Squares Formula

Apply the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\), where \(a = (x-y)^2\) and \(b = 2(x-y)^2\). Thus the factored form will be \((a-b)(a+b)\). Substituting back the values of \(a\) and \(b\) we get, \((x-y)^2 - 2(x-y)^2)((x-y)^2 + 2(x-y)^2)\).

Step 3: Simplifying Terms

Simplified form of the above expression finally becomes \(-1(x - y)^2)(3(x - y)^2)\).

Key Concepts

Difference of Squares

When tackling algebraic expressions, understanding the concept of the difference of squares is essential. It's based on the pattern that occurs when you subtract one square number from another. Mathematically, it's expressed as \(a^2 - b^2\), where both \(a\) and \(b\) are real numbers or algebraic expressions.

This formula is actually an application of the identity \(a^2 - b^2 = (a-b)(a+b)\), which is another way of representing two terms that are squared and subtracted from each other. The identity implies that any quadratic expression that fits the \(a^2 - b^2\) pattern can be factored into the product of two binomials, \(a-b\) and \(a+b\). Recognizing this pattern allows us to simplify and factor complicated expressions.

For instance, in the given exercise, we see \( (x-y)^4 - [2(x-y)^2]^2 \), which is a complex expression. However, once you recognize that this is a difference of squares scenario - where \(a = (x-y)^2\) and \(b = 2(x-y)^2\) - it immediately becomes easier to factor.

Algebraic Expressions

Algebraic expressions are made up of terms combined using mathematical operations like addition, subtraction, multiplication, division, powers, and roots. In the case of polynomial factoring, our attention focuses specifically on expressions involving variables raised to whole number exponents and their coefficients. A well-formed algebraic expression should be simplified, without any like terms that can be combined.

An important aspect of manipulating these expressions is to perform operations following the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our example \( (x-y)^4 - 4(x-y)^2 \), we notice that it's composed of two terms involving a variable to an exponent, thus fitting the definition of a polynomial expression.

Why is simplification crucial?

Before you factor or solve an algebraic expression, simplifying it is crucial because it reveals underlying patterns like the difference of squares, which leads to more straightforward solutions and a clearer understanding of the expression's components and how they interact.

Polynomial Factoring

Polynomial factoring is a critical skill within algebra. It involves breaking down a polynomial into a product of simpler polynomials that, when multiplied, give you the original polynomial. This simplifies expressions and solves equations that would otherwise be too difficult to tackle.

To factor a polynomial, you should first look for a greatest common factor (GCF) and factor that out. If the polynomial is a trinomial, you might employ methods like the FOIL (First, Outer, Inner, Last) technique for factoring quadratics, while for more complex polynomials, such as the given exercise, you may need to employ specialized strategies like factoring by grouping, synthetic division, or looking for patterns like the difference of squares.

Applying the difference of squares to the provided exercise leads to \( (x-y)^2 - 2(x-y)^2)( (x-y)^2 + 2(x-y)^2) \), which can then be simplified further. Recognizing when and how to apply these factoring techniques is the cornerstone of solving polynomial equations effectively.