Problem 95 - Introductory Algebra for College Students

Question

Factor using the formula for the sum or difference of two cubes. $$125 x^{3}-64 y^{3}$$

Step-by-Step Solution

Answer

The factored form of \[125 x^{3}-64 y^{3}\] is \[ (5x - 4y)(25x^2 + 20xy +16y^2)\]

Step 1: Identify 'a' and 'b'

First, identify the correct values for 'a' and 'b'. In this case, since we have \(125x^3\) and \(-64y^3\), we can determine that \(a = 5x\) (since \(5^3x^3 = 125x^3\)) and \(b = 4y\) (since \(-4^3y^3 = -64y^3\)).

Step 2: Apply the formula

Next, apply the formula for the difference of two cubes. Using the formula \[a^3 - b^3= (a − b)(a^2+ ab + b^2)\], and inserting our identified 'a' and 'b' values, the factored form of the given difference of cubes is \[(5x - 4y)((5x)^2 + (5x)(4y) + (4y)^2)\].

Step 3: Simplify

Finally, simplify the result obtained. We get \[ (5x - 4y)(25x^2 + 20xy +16y^2)\].

Key Concepts

Difference of Cubes

The concept of factoring using the difference of cubes is an important algebraic technique. Whenever you need to factor expressions like \(a^3 - b^3\), this formula becomes handy. It allows you to break down complex polynomials into simpler multiplying factors. This is particularly useful in simplifying expressions and solving equations.
In the given exercise, we have \(a = 5x\) and \(b = 4y\). This means the expression \(125x^3 - 64y^3\) fits the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). By substituting \(a\) and \(b\) into this formula, it simplifies into smaller, manageable factors, which is especially helpful for solving algebraic problems.

Polynomials

Polynomials are expressions that consist of variables raised to integer powers, combined using addition, subtraction, and multiplication. A polynomial can have constants, variables, and exponents. The key aspect of polynomials is that the exponents are whole numbers.
In this exercise, both given terms \(125x^3\) and \(-64y^3\) are polynomials. Specifically, they are both cubes because they have a power of three. Cubic polynomials like these are often encountered in algebra, and understanding their structure is crucial to using advanced factoring techniques such as the difference of cubes.
By expressing them as cubes \((5x)^3\) and \((4y)^3\), we can apply specific factoring methods. This organized breakdown helps to efficiently simplify and solve polynomial equations.

Factoring Techniques

Factoring techniques are essential tools in algebra that simplify complex expressions or solve equations. They involve rewriting a polynomial as a product of simpler polynomials. This helps in recognizing patterns and applying formulas like the difference of cubes.
Some of the most common factoring techniques include:

• Factoring out the greatest common factor (GCF)
• Factoring by grouping
• Using special formulas for sums and differences of cubes

In this problem, we employed the technique specific to the difference of cubes. The keyword here is identifying suitable values for \(a\) and \(b\), making use of \(a^3 - b^3\) formula. This approach allows the expression to be broken down into simpler terms, \((5x - 4y)(25x^2 + 20xy + 16y^2)\), making it much easier to handle. Understanding and practicing various factoring techniques can greatly enhance one's mathematical agility.