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If the equation aₙ xⁿ + aₙ₋₁xⁿ⁻¹ +.......... + a₁ x = 0, a₁ ≠ 0, n ≥ 2, has a positive root x = α, then the equation n aₙ xⁿ⁻¹ + (n-1)aₙ₋₁xⁿ⁻² +.......... + a₁ = 0 has a positive root, which is

1. greater than α
2. smaller than α
3. greater than or equal to α
4. equal to α

Correct answer: smaller than α

Solution

Let P(x)=a_n x^n+...+a_1 x; then P(0)=0 and P(alpha)=0. The second polynomial is P'(x). By Rolle's theorem P' has a root in (0,alpha), i.e. a positive root smaller than alpha.

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