%5E3-expand.jpeg "(1-2x)^3 Expand")
Expanding (1-2x)³
Expanding a binomial expression raised to a power can be done using the binomial theorem or by repeated multiplication. Let's explore both methods to expand (1-2x)³.
Expanding using the Binomial Theorem
The binomial theorem states that:
(a + b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b² + ... + ⁿCₙ⁻¹abⁿ⁻¹ + bⁿ
Where ⁿCᵣ represents the binomial coefficient, calculated as:
ⁿCᵣ = n! / (r! * (n-r)!)
Applying this to our problem:
(1 - 2x)³ = 1³ + ³C₁1²(-2x)¹ + ³C₂1¹(-2x)² + ³C₃1⁰(-2x)³
Let's calculate the binomial coefficients:
- ³C₁ = 3! / (1! * 2!) = 3
- ³C₂ = 3! / (2! * 1!) = 3
- ³C₃ = 3! / (3! * 0!) = 1
Now, substitute the values back into the equation:
(1 - 2x)³ = 1 + 3(1²)(-2x) + 3(1¹)(-2x)² + 1(-2x)³
Finally, simplify the expression:
(1 - 2x)³ = 1 - 6x + 12x² - 8x³
Expanding by Repeated Multiplication
We can also expand (1-2x)³ by multiplying it out step by step:
(1 - 2x)³ = (1 - 2x)(1 - 2x)(1 - 2x)
First, multiply the first two factors:
(1 - 2x)(1 - 2x) = 1 - 2x - 2x + 4x² = 1 - 4x + 4x²
Now, multiply the result by the remaining factor:
(1 - 4x + 4x²)(1 - 2x) = 1 - 2x - 4x + 8x² + 4x² - 8x³
Finally, combine like terms:
(1 - 2x)³ = 1 - 6x + 12x² - 8x³
Conclusion
Both methods, using the binomial theorem and repeated multiplication, lead to the same expanded form of (1 - 2x)³ which is 1 - 6x + 12x² - 8x³.