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January 2, 2017

Real Roots of a Cubic Polynomial | TOMATO Objective 258

Try this beautiful problem from TOMATO Objective no. 258 based on Real Roots of a Cubic Polynomial.

Problem: Real Roots of a Cubic Polynomial 

Let a,b,c be distinct real numbers. Then the number of real solution of (x-a)^3+(x-b)^3+(x-c)^3=0 is

(A) 1

(B) 2

(C) 3

(D) depends on a,b,c

Solution: Ans: (A)

Let f(x)=(x-a)^3+(x-b)^3+(x-c)^3

=> f'(x)=3(x-a)^2+3(x-b)^2+3(x-c)^2=0 => f'(x)=(x-a)^2+(x-b)^2+(x-c)^2=0 => x=a, x=b, x=c

But it is not possible a quadratic equation has three roots.so, it implies that  f'(x) has no real roots.But  f(x) is a cubic polynomial. And we know a cubic polynomial must have at least one real root ( we know all polynomial curves are continuous ,so it cuts either of the axes at least once).So, number of real root of the given equation is 1 .

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