## 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 = โ(n choose k) * a^(n-k) * b^k**

Where:

**โ**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 = โ(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.