F(x)= (2x-5)^3

and ?

To expand the function \(f(x) = (2x - 5)^3\), you can use the binomial theorem. The binomial theorem states that for any positive integer \(n\) and any real numbers \(a\) and \(b\), the expansion of \( (a + b)^n \) is given by:

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

where \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \(k\) items from a set of \(n\) distinct items. The binomial coefficient can be calculated using the formula:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

where \(n!\) denotes the factorial of \(n\), which is the product of all positive integers up to \(n\).

In the case of \(f(x) = (2x - 5)^3\), we have \(a = 2x\) and \(b = -5\), and we substitute these values into the binomial theorem formula:

\[
(2x - 5)^3 = \binom{3}{0} (2x)^3 (-5)^0 + \binom{3}{1} (2x)^2 (-5)^1 + \binom{3}{2} (2x)^1 (-5)^2 + \binom{3}{3} (2x)^0 (-5)^3
\]

Simplifying each term, we get:

\[
(2x - 5)^3 = (8x^3) + (-60x^2) + (150x) + (-125)
\]

So, the expansion of \(f(x) = (2x - 5)^3\) is \(8x^3 - 60x^2 + 150x - 125\).