Problem 87 - Introductory Algebra for College Students

Question

Factor using the formula for the sum or difference of two cubes. $$x^{3} y^{3}-64$$

Step-by-Step Solution

Answer

The factorized form of the expression \(x^{3} y^{3}-64\) is \((xy - 4)(x^2y^2 + 4xy + 16)\).

Step 1: Identifying Cubes

First, rewrite the expression as a difference of cubes. Here, we can rewrite the expression as \((xy)^3 - 4^3\) to represent it as a difference between two cubes. In this step, identify \(a=xy\) and \(b=4\).

Step 2: Apply difference of cubes formula

Next, apply the difference of cubes formula to factorize the expression. The formula for the difference of cubes is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Substituting the identified values of \(a\) and \(b\) to the formula, we get this operation: \((a - b)(a^2 + ab + b^2)\) = \((xy - 4)((xy)^2 + (xy)*4 + 4^2)\)

Step 3: Solve the formed expressions

Finally, we further solve the formed expression. The expression \(xy^2 + 4xy + 16\) does not simplify any further. So, the factorization of \(x^{3} y^{3}-64\) is \((xy - 4)(x^2y^2 + 4xy + 16)\).

Key Concepts

Difference of Cubes

The **difference of cubes** is a special pattern for factoring expressions involving terms raised to the third power. This can be recognized by expressions of the form \(a^3 - b^3\). Just like in the exercise provided, we can express an algebraic expression as the difference of two perfect cubes. By clearly identifying both of the cube terms, it makes it a breeze to apply the formula correctly and quickly.

• Identification: Determine the values of \(a\) and \(b\) such that \(a^3 = (xy)^3\) and \(b^3 = 4^3\).
• Formula: The formula for the difference of cubes is \((a^3 - b^3 = (a - b)(a^2 + ab + b^2))\).

Notice how the formula breaks down the expression into a binomial factor \((a - b)\) and a trinomial factor \((a^2 + ab + b^2)\). This method not only makes expressions simpler but also highlights patterns that are easy to spot when factoring polynomials.

Algebraic Expressions

**Algebraic expressions** form the backbone of algebra and consist of variables, numbers, and operations. Understanding how to manipulate these expressions is crucial for simplifying and solving equations.An expression like \(x^3y^3 - 64\) consists of:

• Variables: \(x\) and \(y\) are the variables, representing values that can change.
• Exponents: These indicate that the variable is being raised to a certain power. Here, we see cubes \((^3)\), signifying each variable is multiplied by itself three times.
• Constants: The number 64 in our expression is a constant, a fixed number that does not change.

Breaking down algebraic expressions by their components aids in recognizing their structure. This clarity is especially important when factoring, allowing us to easily identify potential formulas or patterns that can be used, like the difference of cubes.

Polynomial Factorization

**Polynomial factorization** involves breaking down a polynomial into simpler, "factorable" components, often making calculations significantly more manageable.When factorizing polynomials:

• **Look for Patterns:** Patterns such as the difference of cubes provide direct methods for factorization, as seen in the exercise.
• **Use Known Formulas:** Applying formulas simplifies the process tremendously, converting complex expressions into manageable segments.
• **Check Redundancies:** Always verify if further simplification is possible (e.g., factoring a component again if possible). In our case, \((x^2y^2 + 4xy + 16)\) does not simplify further.

By employing strategic factorization techniques on polynomials, solutions become more clear and easy to compute, enhancing problem-solving efficiency and accuracy.