if f (x) is a polynomial of degree 3 whose roots are p,q,r. FIND f (3) if sum of cubes of roots is 0 and sum of squares of roots and sum of roots are unit digit of 3^2018 and 5^50666666666 respectively?

To find the value of f(3), we first need to find the polynomial f(x) using the given information.

Let's start by using the information about the sum of the cubes of the roots, which is equal to 0.

We know that the sum of the cubes of the roots of a cubic polynomial is given by:
p^3 + q^3 + r^3 = 0

Now, using the identity (p + q + r)^3 = p^3 + q^3 + r^3 + 3(pqr)(p + q + r), we can expand the left side of the equation:
(p + q + r)^3 = 0 + 3(pqr)(p + q + r)

Simplifying further:
(p + q + r)^3 = 3(pqr)(p + q + r)

Since we know that the sum of the cubes of the roots is 0, we can rewrite the equation as:
0 = 3(pqr)(p + q + r)

This equation tells us that either p + q + r = 0 or at least one of the roots, p, q, or r is equal to 0. Since we are given that the sum of squares of roots and the sum of roots have unit digit 3^2018 and 5^50666666666 respectively, it implies that p + q + r ≠ 0 and none of the roots are 0.

Now, let's look at the information about the sum of squares of the roots and the sum of the roots.

We know that the sum of the squares of the roots of a cubic polynomial is given by:
p^2 + q^2 + r^2 = (p + q + r)^2 - 2(pqr)

And we are also given that the sum of squares of roots has the unit digit of 3^2018, which means that:
(p + q + r)^2 - 2(pqr) = unit digit of 3^2018

Similarly, we know that the sum of the roots is given by:
p + q + r = unit digit of 5^50666666666

Now, we can find the values of (p + q + r)^2 and (pqr) using the given information.

Let's start by finding the value of (p + q + r)^2:
(p + q + r)^2 = (unit digit of 5^50666666666)^2

Next, let's find the value of (pqr):
(p + q + r)^2 - 2(pqr) = unit digit of 3^2018

Now, we have two equations:
(p + q + r)^2 = (unit digit of 5^50666666666)^2
(p + q + r)^2 - 2(pqr) = unit digit of 3^2018

By solving these equations, you can find the value of (p + q + r) and using that value, you can easily evaluate f(3) by substituting the roots, p, q, and r, into the polynomial f(x).

Remember that the polynomial f(x) whose roots are p, q, and r will have the form:
f(x) = (x - p)(x - q)(x - r)

Once you find the polynomial f(x), substitute x = 3 to calculate f(3).