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Expanding (x - 2y)⁴
Expanding expressions with exponents can be a bit tricky, especially when dealing with binomials like (x - 2y). This article will guide you through the process of expanding (x - 2y)⁴ using the binomial theorem.
The Binomial Theorem
The binomial theorem provides a general formula for expanding expressions of the form (a + b)ⁿ. It states:
(a + b)ⁿ = ₁C₀ aⁿ + ₁C₁ aⁿ⁻¹b + ₁C₂ aⁿ⁻²b² + ... + ₁Cₙ₋₁ abⁿ⁻¹ + ₁Cₙ bⁿ
Where ₁Cᵣ represents the binomial coefficient, calculated as:
₁Cᵣ = n! / (r! * (n-r)!)
Expanding (x - 2y)⁴
Let's apply the binomial theorem to our expression:
- Identify 'a' and 'b': In this case, a = x and b = -2y.
- Determine 'n': Our exponent is 4, so n = 4.
Now we can plug these values into the binomial theorem formula:
(x - 2y)⁴ = ₄C₀ x⁴ + ₄C₁ x³(-2y) + ₄C₂ x²(-2y)² + ₄C₃ x(-2y)³ + ₄C₄ (-2y)⁴
- Calculate the binomial coefficients:
- ₄C₀ = 4! / (0! * 4!) = 1
- ₄C₁ = 4! / (1! * 3!) = 4
- ₄C₂ = 4! / (2! * 2!) = 6
- ₄C₃ = 4! / (3! * 1!) = 4
- ₄C₄ = 4! / (4! * 0!) = 1
- Substitute the coefficients and simplify:
(x - 2y)⁴ = 1 * x⁴ + 4 * x³(-2y) + 6 * x²(-2y)² + 4 * x(-2y)³ + 1 * (-2y)⁴
(x - 2y)⁴ = x⁴ - 8x³y + 24x²y² - 32xy³ + 16y⁴
Conclusion
Therefore, the expanded form of (x - 2y)⁴ is x⁴ - 8x³y + 24x²y² - 32xy³ + 16y⁴. Understanding the binomial theorem allows us to expand complex expressions efficiently and accurately.