Computing the coefficient in a large polynomial

Question

What is the coefficient of $x^{10}$ in the expansion of $(5 x^2 + 3)^{14}$?

(I would prefer to know the answer as a mathematical expression rather than a number.)

May I know how to approach this question and what will be the answer? Thanks

Answer 1

Apply binomial theorem to solve for the $x^{10}$ component:

$$\binom{14}{5}(5x^2)^{5}3^{14-5} = 123141768750 x^{10}$$

Answer 2

Assuming the question is asking what is the coefficient of the $x^{10}$ term of the polynomial $(5x^2+3)^{14}$.

You can break this up into two parts, the first part is how can you get a $x^{10}$ term, and the second part is how many ways of getting that term are there.

For the first part, the only way to get the $x^{10}$ term using $5x^2$ and $3$ is to have 5 of the $5x^2$ terms and hence 9 of the $3$ term. This results in the term $(5x^2)^5 \cdot 3^9 = 5^5 \cdot 3^9 \cdot x^{10}$

For the second part, how many unique ways can you multiply the terms to get 5 of the $5x^2$ terms and hence 9 of the $3$ term, and using some basic combinatorics you get $ 14\choose 5$.

Hence the final coefficient on the $x^{10}$ term is $14\choose 5$ $5^5 \cdot 3^9 $