%5E3-expanded.jpeg "(x-6)^3 Expanded")
Expanding (x-6)³
In algebra, expanding an expression means writing it out in a simpler form without parentheses. This is useful for solving equations and simplifying expressions. Let's explore how to expand the expression (x-6)³.
Understanding the Concept
The expression (x-6)³ represents the product of (x-6) multiplied by itself three times:
(x-6)³ = (x-6) * (x-6) * (x-6)
Using the Distributive Property
To expand this expression, we can use the distributive property multiple times:
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First Expansion: Multiply the first two factors: (x-6) * (x-6) = x(x-6) - 6(x-6) = x² - 6x - 6x + 36 = x² - 12x + 36
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Second Expansion: Now, multiply the result from step 1 by the third factor: (x² - 12x + 36) * (x-6) = x(x² - 12x + 36) - 6(x² - 12x + 36) = x³ - 12x² + 36x - 6x² + 72x - 216
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Combining Like Terms: Finally, combine the terms with the same powers of x: x³ - 12x² - 6x² + 36x + 72x - 216 = x³ - 18x² + 108x - 216
Alternative Method: The Binomial Theorem
Another approach to expanding this expression is using the Binomial Theorem. This theorem provides a formula for expanding expressions of the form (a+b)ⁿ. In our case, a=x, b=-6, and n=3.
The Binomial Theorem states: (a+b)ⁿ = ∑(k=0 to n) [nCk * a^(n-k) * b^k]
where nCk is the binomial coefficient, calculated as n!/(k!*(n-k)!).
Applying this to our expression:
(x-6)³ = (³C₀ * x³ * (-6)⁰) + (³C₁ * x² * (-6)¹) + (³C₂ * x¹ * (-6)²) + (³C₃ * x⁰ * (-6)³)
Calculating the binomial coefficients and simplifying:
(x-6)³ = (1 * x³ * 1) + (3 * x² * -6) + (3 * x * 36) + (1 * 1 * -216)
(x-6)³ = x³ - 18x² + 108x - 216
Conclusion
We have successfully expanded the expression (x-6)³ using both the distributive property and the Binomial Theorem, arriving at the same result: x³ - 18x² + 108x - 216. Understanding these methods is crucial for simplifying expressions and solving algebraic problems.