How Old Is Willie?

Some time ago I came across an interesting problem on @MathsJam, taken from Trigg’s Mathematical Quickies:

Trigg MQ 236. “Did your teacher give you that problem?” I asked. “It looks rather tedious.”
“No,” said Willie, “I made it up. It’s a polynomial equation with my age as a root. That is, x stands for my age my last birthday.”
“Well, then,” I remarked, “It shouldn’t be so hard to work out– integer coefficients, integral root. Suppose I try x = 7 \dots. No, that gives 77.”
“Do I look only seven years old?” demanded Willie.
“Well, let me try a larger integer…. No, that gives 85, not zero.”
“Oh, stop kidding!” said Willie, looking over my shoulder. “You know I’m older than that.”
How old is Willie?

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