5-binomial-expansion.jpeg "(1-2 X)5 Binomial Expansion")
Expanding the Binomial (1 - 2x)⁵
The binomial theorem provides a powerful tool for expanding expressions of the form (a + b)ⁿ. Let's apply it to expand (1 - 2x)⁵.
The Binomial Theorem
The binomial theorem states:
(a + b)ⁿ = ∑(n choose k) a^(n-k) b^k
Where:
- n is a non-negative integer representing the power.
- k is an integer ranging from 0 to n.
- (n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!).
Expanding (1 - 2x)⁵
Let's substitute a = 1, b = -2x, and n = 5 into the binomial theorem:
(1 - 2x)⁵ = ∑(5 choose k) 1^(5-k) (-2x)^k
Now, let's calculate the terms for each value of k:
- k = 0: (5 choose 0) 1⁵ (-2x)⁰ = 1
- k = 1: (5 choose 1) 1⁴ (-2x)¹ = -10x
- k = 2: (5 choose 2) 1³ (-2x)² = 40x²
- k = 3: (5 choose 3) 1² (-2x)³ = -80x³
- k = 4: (5 choose 4) 1¹ (-2x)⁴ = 80x⁴
- k = 5: (5 choose 5) 1⁰ (-2x)⁵ = -32x⁵
Finally, we combine these terms to obtain the complete expansion:
(1 - 2x)⁵ = 1 - 10x + 40x² - 80x³ + 80x⁴ - 32x⁵
Conclusion
By applying the binomial theorem, we have successfully expanded (1 - 2x)⁵ to its full polynomial form. This expansion is useful for various applications, including calculus, probability, and numerical analysis.