Question

In the open interval (0, 1), the polynomial $P(x) = x^4 - 4x^3 + 2$ has \(\underline{\hspace{1cm}}\).

• A. two real roots
• B. one real root
• C. three real roots
• D. no real roots

Hint:

If a polynomial has no turning points inside an interval, it can cross the x-axis at most once there.

Answer 1

The Correct Option is B

Consider $P(x) = x^4 - 4x^3 + 2$.
Evaluate at endpoints of (0, 1):
$P(0) = 2 > 0$
$P(1) = 1 - 4 + 2 = -1 < 0$
Since $P(x)$ is continuous, by Intermediate Value Theorem a root lies inside $(0,1)$.
Now check if more than one root exists:
Derivative: $P'(x) = 4x^3 - 12x^2 = 4x^2(x - 3)$.
Critical points inside (0,1): only $x=0$. No turning point inside (0,1).
Thus the function is strictly decreasing on (0,1), so it crosses zero only once.
Final Answer: One real root