Answer:
To simplify the expression (1+x)^4 - 1 using the difference of powers, we can expand the binomial (1+x)^4 using the binomial theorem. The binomial theorem states that for any positive integer n, the expansion of (a+b)^n can be written as the sum of the terms of the form (n choose k) * a^(n-k) * b^k, where (n choose k) represents the binomial coefficient.
In this case, a = 1, b = x, and n = 4. So, the expansion of (1+x)^4 is:
(4 choose 0) * 1^4 * x^0 + (4 choose 1) * 1^3 * x^1 + (4 choose 2) * 1^2 * x^2 + (4 choose 3) * 1^1 * x^3 + (4 choose 4) * 1^0 * x^4
Simplifying the terms:
1 * 1 * 1 + 4 * 1 * x + 6 * 1 * x^2 + 4 * 1 * x^3 + 1 * 1 * x^4
1 + 4x + 6x^2 + 4x^3 + x^4
Now, we subtract 1 from this expression:
1 + 4x + 6x^2 + 4x^3 + x^4 - 1
Simplifying further:
4x + 6x^2 + 4x^3 + x^4
So, the simplified form of (1+x)^4 - 1 is 4x + 6x^2 + 4x^3 + x^4.
Explanation: