# Establish all polynomial functions, which have the property f(x)=f'(x)*f"(x), x is in the real number set.

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### 1 Answer

Let's suppose that f(x)'s grade is n. We know that if we multiply 2 polynomials, their grades are adding, in order to find the resulting grade.

If the grade of f(x) is n, then the grade of f'(x) is (n-1) and the grade of f"(x) is (n-2).

n= n-1+n-2, n=3

f(x)= ax^3 + bx^2 +cx+d

f'(x)=3ax^2+2bx+c

f"(x)=6ax+2b

f(x)=f'(x)*f"(x)

ax^3 + bx^2 +cx+d=(3ax^2+2bx+c)(6ax+2b)

The expressions above are eqaul if only the corresponding quotients are equals.

a=18a^2, **a=1/18**

b=18ab=18*(1/18)b, so **b could be any real number**

c=4b^2+6ac, **c=6b^2**

d=2bc, **d=12b^3**

**f(x)= (1/18)x^3 + bx^2 +6bx+12b^3**