%5E3-expand-formula.jpeg "(x-1)^3 Expand Formula")
Understanding the Expansion of (x-1)³
The expression (x-1)³ represents the cube of the binomial (x-1). Expanding this expression means writing it out as a sum of terms. While you can simply multiply (x-1) by itself three times, there's a more efficient way to expand this using the binomial theorem.
The Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ, where n is a non-negative integer. The theorem states:
(a + b)ⁿ = ⁿC₀aⁿb⁰ + ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b² + ... + ⁿCₙa⁰bⁿ
Where ⁿCᵣ represents the binomial coefficient, calculated as:
ⁿCᵣ = n! / (r! * (n-r)!)
Expanding (x-1)³
Let's apply the binomial theorem to expand (x-1)³:
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Identify a and b: In our case, a = x and b = -1.
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Apply the formula: (x-1)³ = ³C₀x³(-1)⁰ + ³C₁x²(-1)¹ + ³C₂x¹(-1)² + ³C₃x⁰(-1)³
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Calculate the binomial coefficients:
- ³C₀ = 3! / (0! * 3!) = 1
- ³C₁ = 3! / (1! * 2!) = 3
- ³C₂ = 3! / (2! * 1!) = 3
- ³C₃ = 3! / (3! * 0!) = 1
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Substitute the coefficients and simplify: (x-1)³ = 1x³1 + 3x²(-1) + 3x1 + 11(-1)
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Final Expansion: (x-1)³ = x³ - 3x² + 3x - 1
Conclusion
Expanding (x-1)³ using the binomial theorem gives us the expression x³ - 3x² + 3x - 1. This method provides a systematic way to expand binomials raised to any power, making it a valuable tool in algebra and other areas of mathematics.