Latest TCS Aptitude Question SOLUTION: How many polynomials of degree >=1 satisfy f(x2)=[f(x)]2=f(f(x) a) more than 2 b) 2 c) 0 d) 1

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How many polynomials of degree >=1 satisfy f(x2)=[f(x)]2=f(f(x)
a) more than 2
b) 2
c) 0
d) 1

Asked In TCS (9 years ago)

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Read Solution (2) :

shravya  (9 years ago)

let f(x) be a polynomial of degree n>=1
then comparing the degrees of f(x^2) , (f(x))^2 and f(f(x)) we have 2n=2n=n^2
since n not = 0 we have n=2
so the polynomial f(x) be ax^2+bx+c where a not = 0
then putting this in f(x^2)=(f(x))^2=f(f(x)) and expanding gives
ax^4+bx^2+c = a^2x^4+2abx^3+(2ac+b^2)x^2+2bcx+c^2=a^3x...
comparing the co-efficients of f(x^2) , (f(x))^2 and f(f(x)) respectively
we have
a=a^2=a^3
0=2ab=2a^2b
b=2ac+b^2=ab^2+2a^2c+ab
0=2bc=2abc+b^2
c=c^2=bc+c
solving these we get a=1 , b=0, and c=0
therefore the polynomial is f(x)=x^2 this is the only polynomial satisfying the given conditions

Is this solution Helpfull? Yes (1) | No (4)  

Sujithra  (9 years ago)

ans will be only one ans: 1

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