Problem 96 - Introductory Algebra for College Students

Question

Factor using the formula for the sum or difference of two cubes. $$125 x^{3}-y^{3}$$

Step-by-Step Solution

Answer

The factored form of \(125x^{3}-y^{3}\) is \((5x-y)(25x^2 + 5xy + y^2)\)

Step 1: Identify the values of a and b

First, identify the values of a and b. The equation has the form \(a^3 - b^3\) where \(a^3 = 125x^3\) and \(b^3 = y^3\). Therefore, \( a = 5x\) and \( b = y\).

Step 2: Apply the formula

Next, apply the formula for the factorization of the difference of two cubes, which is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Substitute \(a = 5x\) and \(b = y\) into this formula to get: \( (5x-y)((5x)^2 + 5x*y + y^2)\)

Step 3: Simplify the expression

Finally, simplify the expression to get the factored form. This provides the final formula: \( (5x-y)(25x^2 + 5xy + y^2)\)

Key Concepts

Sum of Cubes Formula

When encountering algebraic expressions such as the difference of two cubes, understanding the sum of cubes formula can be very helpful. The sum of cubes formula is an algebraic identity that expresses the sum of two cubed numbers as a product of a binomial and a trinomial. In mathematical terms, the sum of cubes is given by:
\[\begin{equation} a^3 + b^3 = (a + b)(a^2 - ab + b^2)ewline\end{equation}\]This formula can be thought of as a special tool in your algebra toolkit, designed to simplify expressions that resemble the sum of two cubed terms. Although the original problem asked for the difference of two cubes, the sum of cubes formula is its counterpart and is worth exploring to grasp the concept behind these identities fully.
Conversely, the difference of cubes which was used in the exercise is written as:
\[\begin{equation} a^3 - b^3 = (a - b)(a^2 + ab + b^2)ewline\end{equation}\]Knowing both formulas allows students to factor similar algebraic expressions effortlessly, whether they are summing or subtracting.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (like x or y), and arithmetic operators (such as plus and minus signs). In the context of our problem, the expression \[\begin{equation} 125x^3 - y^3 ewline\end{equation}\] is an algebraic expression representing the difference of two cubes.
These expressions can vary greatly in complexity. Simplifying them is a fundamental aspect of algebra, which can be done by performing operations like addition, subtraction, multiplication, and division, as well as applying mathematical properties and formulas like the difference of two cubes.
Understanding how to handle algebraic expressions is crucial for solving equations, simplifying polynomials, and working with functions. The ability to manipulate these expressions is a foundational skill that aids in the exploration of more advanced mathematics.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of smaller polynomials that, when multiplied together, give you the original polynomial. It is essentially the opposite of expanding a polynomial. The goal of factorization is to simplify the polynomial and find its roots (the values of the variable that make the polynomial equal to zero).
For instance, the factored form of the difference of cubes, such as in our original problem, accomplishes this by reducing a cubic expression into a product of a binomial and a trinomial. Mastering polynomial factorization is essential for solving polynomial equations and for understanding more advanced concepts in algebra and calculus.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form while maintaining their original value. This process makes expressions more manageable and prepares them for solving or further manipulation. Simplification can involve combining like terms, factoring, expanding expressions, and canceling common factors in fractions.
In the exercise provided, simplification occurs after applying the difference of cubes formula. To simplify the product of \[\begin{equation} (5x - y) ewline\end{equation}\] and the trinomial, one might look for like terms and common factors. However, as the trinomial is already in its simplest form, no further simplification is needed. Understanding simplification helps in deciphering complex problems and is a vital component of algebraic problem-solving.