Calculus – Proof That a Degree 4 Polynomial Has at Least Two Roots

Question

Let $$P(x) = x^4+a_3x^3+a_2x^2+a_1x+a_0$$
$$P(x_0) = 0$$
$$P'(x_0) \not= 0$$
with $x_0$ and each $a_i$ real. Prove that $P(x)$ has a at least two real roots.

I can't figure why this is true.

Answer 1

Hint: have you tried sketching possible shapes for the polynomial, noting that the $x^4$ term dominates when $|x|$ is large.

Hint: can you show that there is a value of $x$ for which the polynomial takes a negative value? Then use the intermediate value theorem.