Factoring cubic expressions can seem daunting, but with the right approach, it becomes straightforward. This guide will walk you through factorizing x³ - 8 step-by-step, explaining the concepts along the way. We'll focus on using the difference of cubes formula, a powerful tool for tackling this type of problem.
Understanding the Difference of Cubes
The key to factorizing x³ - 8 lies in recognizing it as a difference of cubes. The general formula for the difference of cubes is:
a³ - b³ = (a - b)(a² + ab + b²)
In our case, x³ - 8 can be rewritten as:
x³ - 2³
This clearly fits the difference of cubes formula, where:
- a = x
- b = 2
Applying the Formula
Now, let's substitute these values into the difference of cubes formula:
x³ - 2³ = (x - 2)(x² + 2x + 4)
And there you have it! We've successfully factorized x³ - 8 into (x - 2)(x² + 2x + 4).
Verifying the Factorization
To verify our answer, you can expand the factored expression:
(x - 2)(x² + 2x + 4) = x(x² + 2x + 4) - 2(x² + 2x + 4) = x³ + 2x² + 4x - 2x² - 4x - 8 = x³ - 8
This confirms our factorization is correct.
Why This Method Works
The difference of cubes formula works because of the algebraic properties of expanding brackets. If you were to expand (a - b)(a² + ab + b²), you would arrive back at a³ - b³. This is a fundamental identity in algebra.
Further Applications
This factorization technique is widely applicable in various mathematical contexts, including:
- Calculus: Simplifying expressions before integration or differentiation.
- Algebraic manipulation: Solving cubic equations.
- Geometry: Calculating volumes or areas of 3-dimensional shapes.
Understanding the difference of cubes factorization is a valuable skill for anyone studying algebra and beyond. Remember to practice regularly to solidify your understanding and build confidence in your ability to tackle more complex problems.
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