Problem 42 - Elementary Algebra

Question

Factor \(x^{4}-y^{4}\).

Step-by-Step Solution

Answer

Answer: \((x^{2} + y^{2})(x + y)(x - y)\)

Step 1: Identify as Difference of Squares

Recognize that \(x^{4}-y^{4}\) can be written as a difference of squares: \((x^{2})^2-(y^{2})^2\).

Step 2: Apply the Difference of Squares Formula

Apply the difference of squares formula to \((x^{2})^2-(y^{2})^2\). This will give us: \(((x^{2})+(y^{2}))(x^{2}-y^{2})\).

Step 3: Apply the Difference of Squares Formula Again

Notice that \(x^{2}-y^{2}\) is also a difference of squares, so apply the difference of squares formula to get \((x+y)(x-y)\).

Step 4: Write the Fully Factored Expression

Combine the factors from Steps 2 and 3: \(((x^{2})+(y^{2}))(x+y)(x-y)\). This is the fully factored form of the given expression.

Key Concepts

Difference of Squares

The concept of the "Difference of Squares" is key in solving many algebraic expressions. It refers to a situation where you have two squares being subtracted from each other. In mathematical terms, it takes the form of \[ a^2 - b^2. \] Recognizing this pattern is extremely useful because it allows you to factor these expressions quickly and easily.
For example, with an expression like \( x^4 - y^4 \), you can see it as \((x^2)^2 - (y^2)^2\), resembling the pattern of a difference of squares. Once identified, you apply the difference of squares formula:

• \( a^2 - b^2 = (a-b)(a+b) \)
• Here, \(a\) is \( x^2\) and \(b\) is \( y^2\).

This formula tells us that every difference of squares can be rewritten as the product of \((a - b)\) and \((a + b)\). By employing this technique, we can break down more complex expressions into simpler factors.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of numbers, variables, and arithmetic operations. Think of them as mathematical phrases. For example, \( x^4 - y^4 \) is an algebraic expression involving variables \( x \) and \( y \), both raised to a power.
These expressions can be manipulated in different ways, often with the goal of simplifying or factoring them. Factoring reduces an expression into a product of simpler expressions, which can make solving equations easier.

• Variables, like \(x\) and \(y\), serve as placeholders for numbers and allow a general representation of mathematical relationships.
• Exponents, such as those in \(x^4\), indicate repeated multiplication.
• Operations, including addition, subtraction, and multiplication, are used to combine variables and numbers.

Understanding these elements helps in reshuffling or rearranging terms within the expression, as seen when factoring \(x^4 - y^4\) using the difference of squares.

Factoring Techniques

Factoring techniques are methods used to rewrite algebraic expressions as a product of simpler factors. These techniques are crucial for solving equations because they often reveal solutions more straightforwardly than solving the original form of an expression.
Some common factoring techniques include:

• GCF (Greatest Common Factor): Find the largest number or variable common to all terms.
• Difference of Squares: Recognize expressions that fit the pattern \( a^2 - b^2 \) and apply the formula \((a-b)(a+b)\).
• Trinomials: Factor expressions like \( ax^2 + bx + c \) into two binomials.

For the expression \(x^4 - y^4\), using the difference of squares technique simplifies it to its factors quickly:

• First, we identify it as \((x^2)^2 - (y^2)^2\), then apply the difference of squares.
• Next, recognizing that one of the resulting factors, \(x^2 - y^2\), is also a difference of squares permits further factoring into \((x - y)(x + y)\).

This step-by-step approach illustrates how understanding and applying proper factoring techniques can lead to a completely factored solution.