%5E4.jpeg "(x-6)^4")
Exploring the Expansion of (x-6)^4
The expression (x-6)^4 represents the product of (x-6) multiplied by itself four times:
(x-6) * (x-6) * (x-6) * (x-6)
While we could manually multiply this out, it's more efficient to use the binomial theorem.
Understanding 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 formula is:
(a + b)^n = ∑(n choose k) * a^(n-k) * b^k
Where:
- ∑ represents the sum over all values of k from 0 to n.
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying the Binomial Theorem to (x-6)^4
- Identify 'a' and 'b': In our case, a = x and b = -6.
- Determine 'n': n = 4.
Now, we can use the binomial theorem to expand (x-6)^4:
(x - 6)^4 = ∑(4 choose k) * x^(4-k) * (-6)^k
Let's break this down for each value of k:
- k = 0: (4 choose 0) * x^4 * (-6)^0 = 1 * x^4 * 1 = x^4
- k = 1: (4 choose 1) * x^3 * (-6)^1 = 4 * x^3 * -6 = -24x^3
- k = 2: (4 choose 2) * x^2 * (-6)^2 = 6 * x^2 * 36 = 216x^2
- k = 3: (4 choose 3) * x^1 * (-6)^3 = 4 * x * -216 = -864x
- k = 4: (4 choose 4) * x^0 * (-6)^4 = 1 * 1 * 1296 = 1296
Finally, we sum up all these terms:
(x-6)^4 = x^4 - 24x^3 + 216x^2 - 864x + 1296
Conclusion
The expansion of (x-6)^4 using the binomial theorem gives us a polynomial expression with five terms: x^4 - 24x^3 + 216x^2 - 864x + 1296. This method provides a structured and efficient way to expand any binomial raised to a non-negative integer power.