How many real roots does the equation $x ^ { 5 } - 7 x + c = 0$ have in the interval $[ - 1,1 ]$ ?
A) at most five real roots
B) no real roots
C) at most one real root
D) at most two real roots
E) at most three real roots

Question

Quizplus · Accepted Answer

Since $x^5$ and $7x$ have opposite signs in the interval $[-1,1]$, the polynomial changes signs at least three times. Therefore, by Descartes’ Rule of Signs, the equation has at most one positive root and at most one negative root. Since the sum of these roots is odd, there must be at least one real root in the interval $[-1,1]$. Thus, the equation has at most one real root in this interval. Therefore, the answer is $\boxed{	extbf{(C)}}$.

How Many Real Roots Does the Equation x5−7x+c=0x ^ { 5 } - 7 x + c = 0x5−7x+c=0

How Many Real Roots Does the Equation $x ^ { 5 } - 7 x + c = 0$