Find the sum of the coefficients of the even powers in the polynomial that results from the expression $f(x)=(x^{3} - x + 1)^{100}$ after expanding the brackets and combining like terms.
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What will be the result if we substitute $x=1$ and $x=-1$ into the given expression?

## Solution

If we substitute $x=1$ into the polynomial $f(x)$, we get the sum of all coefficients of $x^{k}$. If we substitute $x=-1$, we get the difference between the sums of coefficients of even and odd powers. Therefore, the sum of the coefficients of even powers is $1 / 2(f(1)+f(-1))=1$.

## Answer

1.