Problem 42 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ x^{8}-y^{2} $$

Step-by-Step Solution

Answer

Question: Factor the binomial \(x^8 - y^2\). Answer: \((x^4 + y)(x^4 - y)\)

Step 1: Identify the difference of squares

The given binomial is \(x^8 - y^2\). We have two square terms, \(x^8 = (x^4)^2\) and \(y^2 = (y)^2\). As a result, we can express the given binomial as a difference of squares, written as \((a^2 - b^2)\), where \(a = x^4\) and \(b = y\).

Step 2: Factor the difference of squares

Now that we have identified the binomial as \((a^2 - b^2)\), we can proceed to factor it using the difference of squares formula: \((a^2 - b^2) = (a + b)(a - b)\). In this case, \(a = x^4\) and \(b = y\), so we get: $$(x^8 - y^2) = (x^4 + y)(x^4 - y)$$ The factored form of the given binomial is: \((x^4 + y)(x^4 - y)\).

Key Concepts

Binomials

A binomial is a type of algebraic expression that consists of exactly two terms. In the given exercise, the expression is \(x^8 - y^2\). This is a binomial because it has two separate terms: \(x^8\) and \(-y^2\). Often, binomials can be factored or simplified, which allows us to break them down into products of simpler expressions. It's like finding two numbers that multiply together to get a specific result. Understanding how to work with binomials is crucial, as it's a common skill needed for solving many algebraic equations.
Recognizing whether a binomial can be factored often involves identifying special patterns or formulas, such as the difference of squares, which is used here.

Factoring

Factoring is the process of breaking down an expression into simpler "factors" that, when multiplied together, give the original expression. In our exercise, we started with the binomial \(x^8 - y^2\).

Here, we can recognize that each term in the binomial is a perfect square: \(x^8\) can be rewritten as \((x^4)^2\), and \(y^2\) remains \((y)^2\).
One effective method for factoring this type of expression is called the difference of squares. The formula is:

• \((a^2 - b^2) = (a + b)(a - b)\)

With \(a = x^4\) and \(b = y\), applying this formula gives us:

• \((x^4 + y)(x^4 - y)\)

This is the factored form of the original expression. Factoring expressions like these can simplify calculations and is a powerful tool in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition and subtraction. In our exercise, the expression \(x^8 - y^2\) is an example. These expressions can represent real-world situations and are fundamental units in algebra.
An expression becomes especially useful when we can transform it. Factoring is one way to do this.
It's important to understand how algebraic expressions behave because they form the basis for solving equations, understanding functions, and modeling various scenarios.
When working with them, keep in mind:

• Look for recognizable patterns (like squares or cubes of numbers).
• Rewriting terms can often make them easier to manipulate.
• Simplifying expressions might make solving an equation more straightforward.

Algebraic expressions, such as binomials, offer a dynamic way of representing numbers and their relationships. They're like the foundation blocks in the world of algebra.