If f(x)=(x-α)n.g(x) then we always have f(α)=f'(α)=f"(α)=...=fn-1(α) where f(x) and g(x) are polynomial functions.Provided that f(x) has rational coefficients
Q1 If f(x) is of deg 4 and touches x axis at (√3,0).Find sum of roots
Q2 Suppose f(x) touches X-axis at only one point then prove that point of touching must eb always a rational qty
Q3 If f(x) is 3rd degree polynomial and touches Xaxis ,then prove that all roots of f(X) are rational
106Q3. An equation touching the x-axis means that there is a repeated root there. So, atleast 2 real roots are there of f(x) where it tuches x-axis
And as two roots are real and coefficients are rational so all three roots are real
106Q1. 0 ... as x=√3 is a repeated root (repeated twice)
and for rational coefficients we have irrational roots occuring in pairs... So x= -√3 is also a repeated root (twice)
So sum of roots = 0
106Q2. If f(x) touches x-axis at only one point then there is a repeated root.
Further if that root is irrational, then its conjugate (if u know wat i mean) will also be a root... But it is given that it touches exactly at one point.. which is contradictory if there is rational point (where it touches)
So that point has to be real where f(x) touches the x-axis
