2001 SMT/Algebra Problems/Problem 4

Problem

$p(x)$ is a real polynomial of degree at most 3. Suppose there are four distinct solutions to the equation $p(x)$ = 7. What is $p(0)$ ?

Solution

Any value of $p(x)$ can have at most three solutions due to being at most cubic. But there are four distinct solutions, so $p(x)$ must be a constant function; thus $p(0)=p(x)=\boxed{7}$ .

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2001 SMT/Algebra Problems/Problem 4

Problem

Solution