what is (x+h)^3

To expand the expression (x + h)^3, we can use the binomial theorem. The binomial theorem states that for any real numbers a and b and any positive integer n, the expansion of (a + b)^n can be written as:

(a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1} b^1 + \binom{n}{2}a^{n-2} b^2 + \ldots + \binom{n}{n-1}a^1 b^{n-1} + \binom{n}{n}a^0 b^n

where \binom{n}{k} represents the binomial coefficient, and is equal to \frac{n!}{k!(n-k)!}.

In our case, a = x and b = h, so we can substitute these values into the binomial theorem formula:

(x + h)^3 = \binom{3}{0}x^3h^0 + \binom{3}{1}x^2h^1 + \binom{3}{2}x^1h^2 + \binom{3}{3}x^0h^3

Simplifying each term:

\binom{3}{0}x^3h^0 = 1x^3h^0 = x^3

\binom{3}{1}x^2h^1 = 3x^2h^1 = 3x^2h

\binom{3}{2}x^1h^2 = 3x^1h^2 = 3xh^2

\binom{3}{3}x^0h^3 = 1x^0h^3 = h^3

Now we can combine all the terms:

(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3

Therefore, the expansion of (x + h)^3 is x^3 + 3x^2h + 3xh^2 + h^3.

Answer: (x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3