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Polynomials : How to obtain a Polynomial From Finitely Many Points on the Polynomial

Alice: "I'm thinking of a polynomial f(x) with non-negative integer coefficients. Can you tell which one?"

Bob: "Well, I need some information."

Alice: "You can pick any real number r and I'll tell you f(r). Um....that is, I'll tell you finitely many digits of f(r) - but as many as you want."

Bob: "Gee - just one value of your polynomial? I dunno...."

Help Bob out. Tell him a good choice for r, and then how he can determine f(x) in finitely many steps.

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Mathematics, Algebra

Polynomials :

"I'm thinking of a polynomial f(x) with non-negative integer coefficients. Can you tell which one?"

You can pick any real number r and I'll tell you f(r). Um....that is, I'll tell you finitely many digits of f(r) - but as many as you want."

A good choice for r, and then how he can determine f(x) in finitely many steps.

Solution :

Given any real number r , you would be able to tell f(r) - value for any f(x) , just by using substitution method .

Let's assume f(x) as those given below and now calculate the f(r) for any given r-value.

Example 1 : Consider a linear function : f(x) = 2x + 3

When x = 2 , f(2) = 2(2) + 3 = 7
When x = 4 , f(4) = 2(4) ...

Solution Summary

The method of finding a polynomial using points on the polynomial is discussed in detail. The solution is detailed and well presented.

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