%5E5-expand.jpeg "(x-y)^5 Expand")
Expanding (x-y)^5
Expanding expressions like (x-y)^5 can be a tedious process if done manually. However, there are two main methods we can use to simplify this:
1. Binomial Theorem
The binomial theorem provides a general formula to expand expressions of the form (x+y)^n. For (x-y)^5, the formula becomes:
(x-y)^5 = ⁵C₀x⁵(-y)⁰ + ⁵C₁x⁴(-y)¹ + ⁵C₂x³(-y)² + ⁵C₃x²(-y)³ + ⁵C₄x¹(-y)⁴ + ⁵C₅x⁰(-y)⁵
Where ⁵Cₙ represents the binomial coefficient, calculated as ⁵Cₙ = 5! / (n! * (5-n)!).
Let's calculate the binomial coefficients and expand the expression:
- ⁵C₀ = 5! / (0! * 5!) = 1
- ⁵C₁ = 5! / (1! * 4!) = 5
- ⁵C₂ = 5! / (2! * 3!) = 10
- ⁵C₃ = 5! / (3! * 2!) = 10
- ⁵C₄ = 5! / (4! * 1!) = 5
- ⁵C₅ = 5! / (5! * 0!) = 1
Therefore, the expansion becomes:
(x-y)⁵ = x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵
2. Repeated Multiplication
We can also expand (x-y)⁵ by repeatedly multiplying (x-y) by itself:
(x-y)⁵ = (x-y) * (x-y) * (x-y) * (x-y) * (x-y)
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Expand the first two terms: (x-y) * (x-y) = x² - xy - xy + y² = x² - 2xy + y²
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Multiply the result by (x-y): (x² - 2xy + y²) * (x-y) = x³ - 2x²y + xy² - x²y + 2xy² - y³ = x³ - 3x²y + 3xy² - y³
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Continue multiplying by (x-y) until you reach the fifth power: (x³ - 3x²y + 3xy² - y³) * (x-y) = x⁴ - 3x³y + 3x²y² - xy³ - x³y + 3x²y² - 3xy³ + y⁴ = x⁴ - 4x³y + 6x²y² - 4xy³ + y⁴ (x⁴ - 4x³y + 6x²y² - 4xy³ + y⁴) * (x-y) = x⁵ - 4x⁴y + 6x³y² - 4x²y³ + xy⁴ - x⁴y + 4x³y² - 6x²y³ + 4xy⁴ - y⁵ = x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵
As you can see, both methods lead to the same result: (x-y)⁵ = x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵
Conclusion
Expanding expressions like (x-y)⁵ can be done using the binomial theorem or repeated multiplication. While the binomial theorem offers a more concise solution, both methods are valid and can be used to arrive at the same expanded form.