2022/23 AHSMC Part 1 Problem 10

Problem 10 of 16

Let $p(x) = 3x^4 + ax^2 + b$ where $a$ , $b$ are real numbers. Suppose that $p(x)$ has exactly three distinct real roots $r$ , $s$ and $t$ with $r < s < t$ . Which one of the following statements must be true?

Show solution

Solution

Observe that if $k$ is a root of $p(x)$ then $p(-k) = p(k) = 0$ implying that $-k$ is also a root of $p(x)$ . Thus we have $r = -t$ , $s = 0$ and so $r < 0 < t$ . This implies $s + t > 0$ , $rt < 0$ , $r + s < 0$ and $r + t = 0$ . The answer is (A).

University of Alberta

Answer choices

Solution