Sol. By the algebraic interpretation of Rolle's theorem, we can say that between any two roots of a polynomial there is always a root of its derivative. Thus, if α is repeated root of a polynomial f(x), then there must be a root of f ′ (x) in the interval ( α, α ). This means that α is a root of f ′ (x) = 0

∴ f ′ ( α ) = 0

Thus, if α is a repeated root of a polynomial f(x) = 0, then f( α ) = 0 and f ′ ( α ) = 0.

If possible, let φ (x) =1+ + +........+

have a repeated root α .

Then,

φ ( α ) = 0 and φ′ ( α ) = 0

⇒ 1 + + +.....+ = 0 and 1 + α + + ......+ = 0

⇒ = 0 [Subtracting the two equations]

⇒ α = 0

Thus, 0 is a repeated root of φ (x) = 0

But, 0 does not satisfy φ (x) = 0 i.e. it is not a root of φ (x) = 0.

Hence, 1 + + + ..........+ = 0 cannot have a repeated root.