Problem 55 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ x^{12}-y^{12} $$

Step-by-Step Solution

Answer

$$ Answer: The factored form of the expression $$x^{12} - y^{12}$$ is $(x^6 + y^6)((x^3 + y^3)(x^3 - y^3))$.

Step 1: Identify the Difference of Squares

Given the expression $$x^{12} - y^{12},$$ we can rewrite it as $$(x^6)^2 - (y^6)^2$$ to highlight that it is a difference of squares.

Step 2: Apply the Difference of Squares Formula

Using the formula $$a^2 - b^2 = (a + b)(a - b),$$ we substitute \(a = x^6\) and \(b = y^6\): $$ (x^6)^2 - (y^6)^2 = (x^6 + y^6)(x^6 - y^6) $$

Step 3: Further Factor if Possible

Now, we see that the expression \(x^6 - y^6\) is also a difference of squares. We can rewrite it as $$(x^3)^2 - (y^3)^2$$ and apply the difference of squares formula again: $$ (x^3)^2 - (y^3)^2 = (x^3 + y^3)(x^3 - y^3) $$ Finally, we put everything back together: $$ x^{12} - y^{12} = (x^6 + y^6)((x^3 + y^3)(x^3 - y^3)) $$

Key Concepts

Difference of Squares

The concept of the difference of squares is a fundamental idea in algebra. It refers to a binomial expression that takes the form of \( a^2 - b^2 \). This concept is crucial because it can be factored into two simpler binomial expressions through a special factoring technique. The formula used is \( a^2 - b^2 = (a + b)(a - b) \). This allows us to break down complex expressions into simpler parts.

In our example problem, \( x^{12} - y^{12} \), we first express each term as a square: \( (x^6)^2 - (y^6)^2 \). Recognizing this as a difference of squares, we can apply our formula:

• \( (x^6)^2 - (y^6)^2 = (x^6 + y^6)(x^6 - y^6) \)

By understanding and applying the difference of squares technique, you can simplify many algebraic expressions that might appear complex at first glance. This method is not only applicable here but is a versatile tool in many algebraic problems.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operators. They form the basis of algebra and are used to represent mathematical ideas in a symbolic form. Each part of an expression holds specific meaning and follows certain rules.

In our example, the expression \( x^{12} - y^{12} \) comprises two terms: \( x^{12} \) and \( y^{12} \). Both terms involve variables raised to a power, indicating repeated multiplication. Such expressions can often be simplified or manipulated using various algebraic rules and techniques—factoring is one prominent technique.

Understanding the structure and components of algebraic expressions makes them easier to manipulate, reduce, and solve. This foundational knowledge enables you to tackle more complex problems by breaking them down into comprehensible parts.

Factoring Techniques

Factoring is a powerful technique used to simplify algebraic expressions or solve equations. When you factor an expression, you express it as a product of simpler expressions, or factors, which when multiplied together give the original expression back.

In our exercise, the expression \( x^{12} - y^{12} \) is factored by recognizing it as a difference of squares first. The goal is to keep breaking it down using known identities or techniques until it cannot be factored further with ease:

• First factor: \( (x^6 + y^6)(x^6 - y^6) \)
• Further breaking down: \( (x^3 + y^3)(x^3 - y^3) \) from the second term \( x^6 - y^6 \)

Factorization not only simplifies expressions but also reveals their essential structure. This technique is useful in solving equations, simplifying expressions, and analyzing mathematical problems in algebra. Each successful step in factoring brings us closer to revealing the expression's components, making it more manageable and easier to interpret.