How To Factor X 3 125

How to Factor x³ + 125: A Comprehensive Guide

Factoring cubic expressions can seem daunting, but with a systematic approach, it becomes manageable. This comprehensive guide will walk you through the process of factoring x³ + 125, explaining the underlying principles and providing you with the tools to tackle similar problems. We'll explore the concept of the sum of cubes, delve into the factoring formula, and illustrate the process with step-by-step examples. Furthermore, we'll look at how to verify your answer and consider related problems to solidify your understanding.

Understanding the Sum of Cubes

The expression x³ + 125 is a perfect example of a sum of cubes. A sum of cubes is a mathematical expression that can be written in the form a³ + b³. In our case, a = x and b = 5 because 5³ = 125. Recognizing this structure is the key to factoring this expression efficiently.

Identifying a³ and b³

Before applying the formula, it's crucial to correctly identify 'a' and 'b'. In x³ + 125:

• a³ = x³, therefore a = x
• b³ = 125, therefore b = 5 (since 5 x 5 x 5 = 125)

This identification allows us to directly apply the sum of cubes factoring formula.

The Sum of Cubes Factoring Formula

The sum of cubes formula provides a straightforward method for factoring expressions of the form a³ + b³. The formula is:

a³ + b³ = (a + b)(a² - ab + b²)

This formula states that the sum of two cubes can be factored into the product of a binomial (a + b) and a trinomial (a² - ab + b²). Understanding and applying this formula is the core of solving our problem.

Factoring x³ + 125 Step-by-Step

Now, let's apply the sum of cubes formula to factor x³ + 125. Remember we've already identified a = x and b = 5.

1. Substitute 'a' and 'b' into the formula:

   (a + b)(a² - ab + b²) becomes (x + 5)(x² - 5x + 25)

2. Simplify:

   The expression is now factored completely. There are no further common factors to extract.

Therefore, the factored form of x³ + 125 is (x + 5)(x² - 5x + 25).

Verifying Your Answer

It's always a good practice to verify your factoring. You can do this by expanding the factored form:

(x + 5)(x² - 5x + 25)

Using the distributive property (FOIL method), we multiply each term in the first parentheses by each term in the second parentheses:

• x(x²) = x³
• x(-5x) = -5x²
• x(25) = 25x
• 5(x²) = 5x²
• 5(-5x) = -25x
• 5(25) = 125

Combining like terms:

x³ - 5x² + 25x + 5x² - 25x + 125

Notice that the -5x² and 5x² cancel each other out, as do the 25x and -25x. This leaves us with:

x³ + 125

This confirms that our factoring is correct.

Tackling Similar Problems: Variations and Extensions

Understanding the sum of cubes allows you to tackle a range of similar problems. Let's explore a few variations:

1. Factoring with Coefficients:

Consider the expression 8x³ + 27. This is also a sum of cubes, but with coefficients:

• a³ = 8x³ = (2x)³, therefore a = 2x
• b³ = 27 = 3³, therefore b = 3

Applying the formula:

(a + b)(a² - ab + b²) = (2x + 3)((2x)² - (2x)(3) + 3²) = (2x + 3)(4x² - 6x + 9)

2. Factoring a Difference of Cubes

While we focused on the sum of cubes, there's also a formula for the difference of cubes (a³ - b³). The formula is:

a³ - b³ = (a - b)(a² + ab + b²)

For example, factoring x³ - 64 (where a = x and b = 4):

(x - 4)(x² + 4x + 16)

3. Factoring Expressions Requiring Preliminary Simplification

Sometimes, you may encounter expressions that require simplification before applying the sum of cubes formula. For instance:

27x³ + 343y³

Here, a = 3x and b = 7y. Following the formula:

(3x + 7y)((3x)² - (3x)(7y) + (7y)²) = (3x + 7y)(9x² - 21xy + 49y²)

Advanced Applications and Real-World Use Cases

While the immediate application of factoring x³ + 125 might seem limited, the underlying principles of factoring cubic expressions are fundamental in various mathematical and scientific fields. These include:

• Calculus: Finding derivatives and integrals often involves factoring expressions. Understanding factoring techniques helps simplify complex calculations.
• Algebraic Geometry: Factoring is a crucial step in solving polynomial equations, which are essential in defining curves and surfaces.
• Engineering: Many engineering problems involve solving polynomial equations, where factoring can simplify the solution process.
• Physics: Equations describing physical phenomena, such as projectile motion or wave behavior, often involve polynomials that require factoring.

Conclusion

Factoring x³ + 125, and cubic expressions in general, relies on understanding and applying the sum (or difference) of cubes formulas. By systematically identifying 'a' and 'b', substituting them into the appropriate formula, and verifying your answer, you can confidently tackle these problems. This skill is not just a tool for solving specific algebraic exercises, but a foundational concept with broad applications in various areas of mathematics and beyond. Mastering this technique will significantly enhance your mathematical abilities and problem-solving skills. Remember to practice consistently to build proficiency and confidence in your factoring abilities.