Let p(x) be a polynomial with real co-efficients, p(0) = 1 and p'(x) > 0 for all x ∈ ℝ. Then 

Concept:

Monotonic Polynomial Functions:

• A function p(x) is said to be strictly increasing if p′(x) > 0 for all x ∈ ℝ.
• This means that as x increases, the value of p(x) always increases — the graph never turns down.
• Since p(x) is a polynomial with real coefficients, it is continuous and differentiable for all real x.
• Given that p(0) = 1 and p′(x) > 0 ∀ x ∈ ℝ, the function increases from left to right, and 1 is the value at x = 0.
• This implies that for x < 0, p(x) < 1, and for x > 0, p(x) > 1.

Calculation:

Given,

p(0) = 1 and p′(x) > 0 ∀ x ∈ ℝ

⇒ p(x) is strictly increasing on ℝ

⇒ For x < 0, p(x) < 1

⇒ For x > 0, p(x) > 1

⇒ Since p is continuous and increases from p(x) → −∞ to p(0) = 1, it may attain value 0 for some x < 0

⇒ So, p(x) = 0 may have a solution for some negative x

∴ p(x) may have a negative real root.