HMMT February

Algebra Difficulty 4.5 AIME Find the answer

Calculate the sum of the coefficients of $P(x)$ if $\left(20 x^{27}+2 x^{2}+1\right) P(x)=2001 x^{2001}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The sum of coefficients of $f(x)$ is the value of $f(1)$ for any polynomial $f$ . Plugging in 1 to the above equation, $P(1)=\frac{2001}{23}=87$ .

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