%5E3-binomial-expansion.jpeg "(3x-2y)^3 Binomial Expansion")
Understanding Binomial Expansion: (3x - 2y)³
The binomial theorem is a powerful tool used to expand expressions of the form (a + b)ⁿ, where n is a positive integer. Let's explore how it works for the specific case of (3x - 2y)³.
The Binomial Theorem
The binomial theorem states that:
(a + b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + ... + ⁿCₙ₋₁abⁿ⁻¹ + bⁿ
Where ⁿCᵣ represents the binomial coefficient, calculated as:
ⁿCᵣ = n! / (r! * (n-r)!)
Applying the Theorem to (3x - 2y)³
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Identify a and b: In this case, a = 3x and b = -2y.
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Apply the formula:
(3x - 2y)³ = (3x)³ + ³C₁(3x)²(-2y) + ³C₂(3x)(-2y)² + (-2y)³
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Calculate the binomial coefficients:
- ³C₁ = 3! / (1! * 2!) = 3
- ³C₂ = 3! / (2! * 1!) = 3
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Simplify the expression:
(3x - 2y)³ = 27x³ + 3(9x²)(-2y) + 3(3x)(4y²) - 8y³
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Expand and combine terms:
(3x - 2y)³ = 27x³ - 54x²y + 36xy² - 8y³
Conclusion
The binomial theorem provides a systematic way to expand expressions like (3x - 2y)³. By understanding the formula and applying it step-by-step, we can efficiently expand any binomial expression to any power.