%5E4-binomial-expansion.jpeg "(1-2x)^4 Binomial Expansion")
Understanding the Binomial Expansion of (1-2x)^4
The binomial theorem provides a powerful tool for expanding expressions of the form (a + b)^n, where a and b are any real numbers and n is a positive integer. In this case, we'll focus on expanding the expression (1 - 2x)^4.
The Binomial Theorem
The binomial theorem states:
(a + b)^n = (n choose 0)a^n b^0 + (n choose 1)a^(n-1) b^1 + (n choose 2)a^(n-2) b^2 + ... + (n choose n)a^0 b^n
where (n choose k) represents the binomial coefficient, which can be calculated as:
(n choose k) = n! / (k! * (n-k)!)
Expanding (1 - 2x)^4
Applying the binomial theorem, we can expand (1 - 2x)^4 as follows:
(1 - 2x)^4 = (4 choose 0)1^4 (-2x)^0 + (4 choose 1)1^3 (-2x)^1 + (4 choose 2)1^2 (-2x)^2 + (4 choose 3)1^1 (-2x)^3 + (4 choose 4)1^0 (-2x)^4
Now, let's calculate the binomial coefficients:
- (4 choose 0) = 4! / (0! * 4!) = 1
- (4 choose 1) = 4! / (1! * 3!) = 4
- (4 choose 2) = 4! / (2! * 2!) = 6
- (4 choose 3) = 4! / (3! * 1!) = 4
- (4 choose 4) = 4! / (4! * 0!) = 1
Substituting these values back into the expansion:
(1 - 2x)^4 = 1(1)(1) + 4(1)(-2x) + 6(1)(4x^2) + 4(1)(-8x^3) + 1(1)(16x^4)
Simplifying the expression:
**(1 - 2x)^4 = ** 1 - 8x + 24x^2 - 32x^3 + 16x^4
Key Takeaways
- The binomial theorem provides a systematic way to expand expressions in the form (a + b)^n.
- The binomial coefficients determine the numerical factors in each term of the expansion.
- Understanding the binomial theorem allows for the efficient expansion of complex expressions, simplifying calculations and analysis.