Problem 5

Find all polynomials

P(x)

with integer coefficients such that for all real numbers

s

and

t

, if

P(s)

and

P(t)

are both integers, then

P(st)

is also an integer.

Step 6 of 8: Match coefficients to kill lower terms

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P(2x)-P(2t)=2^n\big(P(x)-p\big)\ \Longrightarrow\ a_i(2^i-2^n)=0\ (i<n)\ \Longrightarrow\ P(x)=a_nx^n

Detailed analysis

Both sides have degree $n$ in $x$ ; matching leading coefficients ( $a_n2^n$ on the left) shows $P(2x)-P(2t)$ equals $2^n\big(P(x)-p\big)$ exactly. Comparing the coefficient of $x^i$ for each $i<n$ gives $a_i2^i=2^na_i$ , i.e. $a_i(2^i-2^n)=0$ ; since $2^i\ne2^n$ for $i<n$ , this forces $a_i=0$ . Hence $P(x)=a_nx^n$ .