What is the term independent of $x$ in the expansion of $(2x^{-1} + 3x^2)^{12}$?

Question

What is the term independent of $x$ in the expansion of $(2x^{-1} + 3x^2)^{12}$?

I added the answer, is it asking what value is the expansion where x is not the coefficient? i.e the last answer in the picture?

Answer 1

The independent term is the term where the exponent of $x$ is zero. So yes, in this case, it would be the final number, 10 264 320.

Answer 2

The question is asking which term in that expansion is the coefficient of $x^0$, aka the constant coefficient. Which, in this case, it that last term.

However, I would guess that the question is not asking to test your Wolfram Alpha skills, so I would recommend understanding how you could find just that term without doing the full expansion (hint: put all the terms in the expansion in order of powers of $x$, and see how they jump; maybe you could factor something out of the binomial expression and then apply a familiar formula?)

Answer 3

The $k$-th in the binomial expansion of this binomial is $$\binom {12}k2^kx^{-k}3^{12-k}x^{2(12-k)}=\binom {12}k2^k3^{12-k}x^{24-3k},$$ and it is a constant if and only if $k=8$, in which case the coefficient is $$\binom {12}82^8\,3^4=10\,264\,320.$$