Problem 111 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The completely factored form of the given expression is \((x - y - 2)(x - y)(x - y + 2)(x - y)\).

Step 1: Identify the squares

The given expression is \((x-y)^{4}-4(x-y)^{2}\). Here, \((x-y)^4\) and \(4(x-y)^{2}\) are the squares.

Step 2: Apply the formula for difference of squares

Our expression is of the form \(a^2-b^2\) where \(a = (x-y)^2\) and \(b = 2(x-y)\). Therefore, we can factor the expression using the formula, which gives \((a-b)(a+b)\). After applying this formula, we'll get \(((x-y)^2 - 2(x-y))((x-y)^2 + 2(x-y))\).

Step 3: Simplify factors

Now let's simplify each bracket: \(((x-y)^2 - 2(x-y))\) becomes \((x - y - 2)(x - y)\) and \(((x-y)^2 + 2(x-y))\) becomes \((x - y + 2)(x - y)\).

Key Concepts

Difference of Squares

The difference of squares is a fascinating algebraic identity that helps us factor certain types of expressions. It applies to expressions that can be written as the difference between two perfect squares. The general form for this is \(a^2 - b^2\). This expression can be factored into the product \((a - b)(a + b)\).
This is incredibly useful when trying to simplify expressions without expanding them entirely again. In the original exercise given, we used the property of difference of squares because \((x-y)^4\) is indeed a square, as it equals \((x-y)^2 \times (x-y)^2\), and \(4(x-y)^2\) is also a square \((2(x-y))^2\). This identification allowed us to use the formula to break down the expression into a much simpler form that can be further simplified.
When recognizing a difference of squares, always ensure your terms are indeed squares. Find the root of each term first. After identifying these, apply \((a - b)(a + b)\) for a streamlined solution, helping you break complex expressions into more manageable pieces.

Exponents

Exponents are a way of representing repeated multiplication, making complex calculations more manageable. For example, \((x-y)^4\) means \((x-y)\times(x-y)\times(x-y)\times(x-y)\), and they are crucial in simplifying algebraic expressions.
In our exercise, the expression \((x-y)^4 - 4(x-y)^2\) involved exponents of 2 and 4. Here, the exponent dictates how many times a base, which in this case is \((x-y)\), is multiplied by itself. Exponents have specific rules that can greatly simplify the process of dealing with complex expressions, such as:

• Multiplication: When multiplying similar bases, add the exponents, \(x^m \times x^n = x^{m+n}\).
• Division: When dividing, subtract the exponents, \(x^m / x^n = x^{m-n}\).
• Power of a Power: Multiply the exponents, \((x^m)^n = x^{m \times n}\).

Recognizing these exponent rules allows us to simplify expressions, especially when factoring as we reduce the larger exponents to simpler terms.

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression easier to work with. This is done by reducing it to its simplest form while preserving the original value. Simplifying can involve factoring, combining like terms, and applying basic algebraic identities like the difference of squares.
In our original problem, after applying the difference of squares, we had two expressions needing further simplification: \(((x-y)^2 - 2(x-y))\) and \(((x-y)^2 + 2(x-y))\). Simplification involves breaking these expressions further down by factoring out common variables or terms.
For example:

• The term \((x - y)^2 - 2(x - y)\) was simplified by recognizing and pulling out \((x-y)\) as a common factor, resulting in \((x-y)((x-y) - 2)\).
• Similarly, \((x - y)^2 + 2(x - y)\) could be simplified to \((x-y)((x-y) + 2)\).

In general, simplify expressions by looking for like terms, factoring, and reducing anything that can be made simpler. This helps to make calculations less complex and more straightforward, keeping your work neat and ensuring your understanding of the fundamental concepts is solid.