Problem 53 - Algebra for College Students

Question

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$x^{3} y^{3}-1$$

Step-by-Step Solution

Answer

The factored form is \((xy - 1)(x^2y^2 + xy + 1)\).

Step 1: Recognize the Form

The expression \( x^3 y^3 - 1 \) matches the difference of cubes pattern, which is generally \( a^3 - b^3 \). Identify \( a \) and \( b \) such that \( (x^3 y^3) - 1 = (a^3) - (b^3) \). Here, let \( a = xy \) and \( b = 1 \).

Step 2: Apply the Difference of Cubes Formula

For any difference of cubes \( a^3 - b^3 \), the factorization formula is \( (a - b)(a^2 + ab + b^2) \). Using \( a = xy \) and \( b = 1 \), substitute into the formula to get the factorization: \((xy - 1)((xy)^2 + (xy)(1) + 1^2)\).

Step 3: Simplify the Factors

Simplify the expression \((xy - 1)((xy)^2 + (xy)(1) + 1)\). Calculate \((xy)^2 = x^2y^2\) and recall that \((1)^2 = 1\). Then the expression becomes \(xy - 1)(x^2y^2 + xy + 1)\).

Step 4: Confirm the Factorization

Verify the factorization by expanding \((xy - 1)(x^2y^2 + xy + 1)\). Ensure that the expansion equals the original expression \(x^3 y^3 - 1\). The expansion should simplify correctly to confirm the factorization.

Key Concepts

Difference of Cubes

In algebra, recognizing patterns can make factoring easier. The difference of cubes is one such pattern. When a polynomial looks like two cubic terms subtracted from one another, you might be able to use this method. The general form for the difference of cubes is:

• \( a^3 - b^3 \)

This pattern allows us to factor the expression into a product of two binomials:

• \( (a - b)(a^2 + ab + b^2) \)

Understanding this pattern is crucial for quickly and accurately breaking down complex polynomials. For example, when seeing \( x^3 y^3 - 1 \), one can identify \( a = xy \) and \( b = 1 \) and use the formula to factor the polynomial.

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. It's about finding the unknown or putting real-world variables into equations and then solving them. In algebra, we use letters to represent variables. This allows us to create general formulas that work in many situations.One of the powerful tools in algebra is recognizing and using patterns. Patterns like the difference of cubes help simplify polynomial expressions. With such tools, complex equations become manageable. The factorization of \( x^3 y^3 - 1 \) demonstrates how algebraic techniques can simplify expressions to their core components.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that, when multiplied together, yield the original polynomial. This is an essential skill in algebra as it can simplify equations and make solving them easier.For polynomials that fit specific patterns, like the difference of cubes, shortcuts and formulas are available. The task is to recognize these patterns in the polynomial, apply the correct formula, and simplify the expressions.

• For example, \( x^3 y^3 - 1 \) can be seen as a difference of cubes.
• We use the formula \( (a - b)(a^2 + ab + b^2) \) with \( a = xy \) and \( b = 1 \).

The exercise of confirming factorization by expanding the factors ensures that solutions are accurate and reinforce the concept that factoring polynomials is not just mechanical but also logical and calculated.