Problem 21 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$x^{4}-y^{10}$$

Step-by-Step Solution

Answer

The factored form of \(x^{4}-y^{10}\) is \((x^{2}+y^{5})(x+y\sqrt{y^{4}})(x-y\sqrt{y^{4}})\)

Step 1: Recognize the Difference of Squares

In the expression \(x^{4}-y^{10}\), we recognize that both terms are perfect squares. \(x^{4}\) is the square of \(x^{2}\) and \(y^{10}\) is the square of \(y^{5}\).

Step 2: Apply the Difference of Squares Formula

Apply the formula \(a^{2}-b^{2}=(a+b)(a-b)\), identify \(a\) as \(x^{2}\) and \(b\) as \(y^{5}\). Put \(a\) and \(b\) into the formula to get \((x^{2}+y^{5})(x^{2}-y^{5})\)

Step 3: Look for Additional Difference of Squares

Inspect the factored form to see if there are any additional differences of squares that can be factored. The term \((x^{2}-y^{5})\) is a difference of squares. We can apply the same formula again and factor it − \((x^{2}+y^{5})(x+y\sqrt{y^{4}})(x-y\sqrt{y^{4}})\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a critical skill in algebra that involves breaking down polynomials into their simplest forms — their factors. For students, understanding how to factor polynomials can simplify solving equations and understanding more complex algebraic concepts.

The process begins with identifying the greatest common factor (GCF) that the terms share. After factoring out the GCF, you may use different methods to continue factoring, depending on the polynomial's form. The difference of two squares, which you encountered in the exercise with the expression \(x^{4}-y^{10}\), is one such method.

Polynomials are factored completely when they're written as a product of linear factors or irreducible quadratic factors. Being a fundamental technique, factoring polynomials plays a vital role in calculus, algebra, and beyond, making it an essential tool for students to master.

Perfect Squares

Perfect squares are numbers or algebraic expressions that are the square of integers or other expressions. For instance, the number 9 is a perfect square because it is the product of 3 times 3, or \(3^{2}\).

In algebra, recognizing perfect squares within polynomials, such as \(x^{4}\) and \(y^{10}\) from your exercise, is crucial when factoring expressions like the difference of two squares. A key to understanding perfect squares is knowing the power rule in exponents, which tells us that \(\left(x^{m}\right)^{n} = x^{mn}\).

Being able to quickly identify perfect squares enables you to apply appropriate techniques for simplifying algebraic expressions, making problem-solving more efficient. Hence, memorizing the first few perfect squares and comprehending how to find them can significantly benefit students.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers using the operations of addition, subtraction, multiplication, division, and exponentiation. In the provided exercise \(x^{4}-y^{10}\), for instance, represents a polynomial—an algebraic expression with multiple terms.

Understanding how to manipulate these expressions, such as by applying the rules of exponents or recognizing patterns like the difference of two squares, is essential in solving algebra problems. Learning to work through these expressions with factorization can offer a clearer insight into how the variables relate and how you can simplify or solve equations.

Although they can initially appear complex, breaking down algebraic expressions into their components based on known patterns can make them much more approachable. This, along with regular practice, helps students in building a strong foundation in algebra.

Binomial Theorem

The binomial theorem is a powerful tool which provides a quick way to expand binomials raised to any power. In its most recognizable form, it expresses the expansion of the power of a binomial, \(\left(a + b\right)^{n}\), as a sum of increasingly higher powers of \(a\) and decreasing powers of \(b\).

Although this theorem isn't applied directly in our simple exercise, where we factor the difference of two squares, it's still a valuable piece of the puzzle. The binomial theorem has roots in the patterns you see when factoring polynomials — each factor in the expression \(x^{4}-y^{10}\) initially looks like a binomial expanded via the theorem.

Understanding the binomial theorem aids in identifying patterns within polynomials and can even help with spotting potential differences of squares. As such, it’s beneficial for students to grasp the theorem not only for polynomial expansion but also to enhance their overall algebraic insight.