If f(x) = (x – α ) n g(x) then we always have f( α ) = f ' ( α )
= f" ( α ) =…….= f n–1 ( α )
Where f(x) and g(x) are polynomial functions provided that f(x) has rational coefficients
(i) If f(x) is of degree 4 and touches x-axis at 
then
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Ans.
(i)
Sol. If f(x) touches x-axis at 
and also touches at 
⇒ roots of f(x) are 
(ii)
Sol. If f(z) touches x-axis at only one irrational point, then f(x) = (x – α ) 2 g(x) where α is irrational
⇒ coefficient of f(x) can not be rational ⇒ If f(x) has rational coefficients then point of touching is also rational
(iii)
Sol. The touching point has to be rational
⇒ the two roots of f(x) = 0 are rational
⇒ third root is also rational
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