Problem 69 - Algebra A Combined Function

Question

Factor. $$ x^{6}-y^{3} $$

Step-by-Step Solution

Answer

The expression is factored as \((x^3 - y^{3/2})(x^3 + y^{3/2})\).

Step 1: Identify the Structure

The expression given is a difference of squares or a similar structure, but with powers: \(x^6 - y^3\). Recognize that it can be viewed as \((x^3)^2 - (y^{3/2})^2\). This suggests applying a similar approach to the difference of squares formula.

Step 2: Apply the Difference of Squares Formula

Recall that the difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). In this case, we set \(a = x^3\) and \(b = y^{3/2}\), and apply the formula: \((x^3)^2 - (y^{3/2})^2 = (x^3 - y^{3/2})(x^3 + y^{3/2})\).

Step 3: Further Factor the Terms

Each of the terms \(x^3 - y^{3/2}\) and \(x^3 + y^{3/2}\) can be broken down if they allow further factorization. However, as simples, \(x^3 - y^{3/2}\) and \(x^3 + y^{3/2}\) are irreducible over the real numbers in factor form.

Step 4: Verify the Factorization

To ensure the factorization is correct, multiply the factors \((x^3 - y^{3/2})(x^3 + y^{3/2})\) back together to check if it equals the original expression \(x^6 - y^3\). This confirms the factorization process is accurate.

Key Concepts

Factorization

Factorization is a key concept in algebra that involves breaking down a complex expression into simpler components, or its 'factors'. These factors, when multiplied together, will give back the original expression. Understanding factorization helps simplify expressions, solve equations easier, and discover particularly useful patterns, like the differences of squares.
One of the most common forms of factorization is through identifying familiar patterns such as quadratics or binomials. In our given exercise, you will notice a pattern similar to the 'difference of squares', although it's technically not a square difference, so observation skills are essential. You break down the expression into products of simpler or well-understood expressions to tackle it more easily in future calculations.

Difference of Squares

The difference of squares is a special pattern that's widely used in algebraic expressions. It follows the formula: \(a^2 - b^2 = (a - b)(a + b)\). This formula tells us that we can factor a difference of squares into two binomial expressions.
By identifying expressions that fit this pattern, you can factor and simplify problems neatly. In the expression \(x^6 - y^3\), we can cleverly rewrite it by understanding the exponents: \((x^3)^2 - (y^{3/2})^2\). This resembles the difference of squares formula, although modified for our specific powers. By recognizing these patterns, factorization becomes much more straightforward.

Polynomial Factoring

Polynomials are expressions made up of variables and constants combined using addition, subtraction, and multiplication. Factoring a polynomial means expressing it as a product of its factors or simpler polynomials.
For instance, when we factor \(x^6 - y^3\), we're looking for parts or units we can multiply together to reconstruct the original product. This involves understanding how to identify patterns within the polynomial and knowing various strategies and formulas, like the difference of squares. These techniques help break down complex expressions into manageable pieces to reduce computation complexity later.

Exponents

Exponents are shorthand for repeated multiplication of the same number. They are vital in algebra, giving expressions compact form and highlighting the structure of compound expressions.
In our specific example, \(x^6 - y^3\), understanding exponents helps us rewrite terms in ways that allow efficient factorization, like rewriting \(x^6\) as \((x^3)^2\) and \(y^3\) as \((y^{3/2})^2\). Recognizing how exponents behave in multiplication, division, and when they can be simplified paves the way for smoother algebraic manipulation.