Let the sum of the coefficients of the polynomial $(4x^2 - 4x + 3)^4(4 + 3x - 3x^2)^2$ be $S$ . Find $\frac{S}{16}$ .

Question

Let the sum of the coefficients of the polynomial $(4x^2 - 4x + 3)^4(4 + 3x - 3x^2)^2$ be $S$ . Find $\frac{S}{16}$ .

• Actually I have absolutely no other idea on how to find this without opening up the brackets, and that will seem to be a very tiring work. I don't know any other method of getting the sum of the coefficients of this type of problems.

Wolfram Alpha gives the expansion of this expression :- https://www.wolframalpha.com/input/?i=%284x%5E2+-+4x+%2B+3%29%5E4%284+%2B+3x+-+3x%5E2%29%5E2 .

That is ok, but here I want a short and an easy answer which can immediately tell me the value of $\frac{S}{16}$. Can anyone help me and give some hint or idea ?

Answer 1

Hint: What do you get when you substitute $x=1$ into a polynomial $p(x)$?

Answer 2

The sum of coefficients of a polynomial $P(x)$ is $P(1)$.