We know that α is a root of f(x) = 0. This means (x – α ) is a factor of f(x). If α is a repeated root of f(x) = 0 then f(x) has a repeated factor (x – α ), i.e., (x – α ) 2 is a factor of f(x). But here f(x) is quadratic

∴ f(x) = λ (x – α ) 2 …(1)

Where λ is a constant.

Let Δ (x) =

Which is of the degree 5 at most and 3 at least.

Clearly, Δ ( α ) = 0 …(2)

Differentiating Δ (x) w.r.t. x,

Δ′ (x)= +

+

because derivatives of constants = 0

=

∴Δ′ ( α ) = = 0…(3)

because R 1 ≡ R 3 . We know that

φ ( α ) = 0 ⇒ (x – α ) is a factor of φ (x) and φ′ ( α ) = 0

⇒ (x – α ) 2 is a factor of φ (x)

∴ from (2) and (3), Δ (x) has a factor (x – α ) 2 . ∴ Δ (x)

= (x – α ) 2 . F(x) = λ (x – α ) 2 . F(x) = f(x).

F(x), using (1) ∴ Δ (x) is divisible by f(x).