Problem 41 - Elementary Algebra

Question

Factor completely. $$ x 4-y 4 $$

Step-by-Step Solution

Answer

The complete factorization is \( (x^2 + y^2)(x + y)(x - y) \).

Step 1: Recognize the Expression Type

The expression given is \( x^4 - y^4 \). This is a difference of two squares, since \( x^4 \) and \( y^4 \) are both perfect squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares can be factored as \( a^2 - b^2 = (a + b)(a - b) \). Here, \( a = x^2 \) and \( b = y^2 \), so the expression factors to \( (x^2 + y^2)(x^2 - y^2) \).

Step 3: Further Factor the Difference of Squares

Notice that \( x^2 - y^2 \) is again a difference of squares. Apply the same formula to factor it: \( x^2 - y^2 = (x + y)(x - y) \).

Step 4: Write the Completely Factored Form

Substitute the result from Step 3 into Step 2, yielding the final factored form: \( (x^2 + y^2)(x + y)(x - y) \).

Key Concepts

Difference of Squares

The difference of squares is a specific type of algebraic expression and is represented by the formula:\[ a^2 - b^2 = (a + b)(a - b) \]This equation tells us that a square of one term, subtracted by a square of another term can be factored into a product of two binomials: one sum and one difference. This concept is incredibly useful when factoring polynomials because it reveals a straightforward way to break down more complex expressions into simpler components.

In our exercise, we start with the expression \( x^4 - y^4 \). Notice that both \( x^4 \) and \( y^4 \) are perfect squares: \( x^4 = (x^2)^2 \) and \( y^4 = (y^2)^2 \). This means our expression can initially be factored using the difference of squares formula by considering \( a = x^2 \) and \( b = y^2 \).

This results in the expression being factored to:

• \( (x^2 + y^2) \)
• \( (x^2 - y^2) \)

Identifying and applying the difference of squares is a key skill that will help you tackle many polynomial expressions with ease.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations (like addition and subtraction) grouped together. In algebra, understanding how to manipulate these expressions is essential, especially when it comes to solving equations or factoring polynomials.

The original expression \( x^4 - y^4 \) is an algebraic expression. It consists of two variables, \( x \) and \( y \), each raised to the fourth power. Being able to simplify or rearrange algebraic expressions like this is a core skill in algebra. By recognizing that the expression is a difference of squares, we can apply our mathematical knowledge to simplify it.

When working with algebraic expressions, keep these points in mind:

• Identify if the expression can be simplified or factored.
• Look for patterns, such as the difference of squares, that suggest easy factorization.
• Use known formulas—like \( a^2 - b^2 = (a + b)(a - b) \)—to transform the expression.

Mastering these techniques allows for greater flexibility in algebra, allowing more complex problems to be solved effectively.

Polynomial Factorization

Polynomial factorization is the process of breaking a polynomial down into a product of simpler polynomials. This process is crucial for solving polynomial equations, understanding polynomial functions, and simplifying expressions.

The expression from the exercise, \( x^4 - y^4 \), after recognizing and applying the difference of squares formula, first reduces to:

• \( (x^2 + y^2)(x^2 - y^2) \)

However, \( x^2 - y^2 \) itself is also a difference of squares and can further be factored into \( (x + y)(x - y) \). This illustrates how polynomial factorization is often a multi-step process, breaking down each factor as far as possible.

Key aspects of polynomial factorization include:

• Identifying the type of polynomial and applicable factorization formulas.
• Applying factorization techniques iteratively until all parts are fully factored.
• Writing the final expression as a complete product of factors, for this problem resulting in: \( (x^2 + y^2)(x + y)(x - y) \).

By understanding and applying these steps, polynomial expressions can be simplified to a form that reveals their roots and simplifies complex calculations.