SOLUTION: Let f(x) a polynomial function whose coefficients are integers. Suppose that r is a real zero of f and that the leading coefficient of f is 1.Use the Rational Zeros Theorem to show
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Question 1189083: Let f(x) a polynomial function whose coefficients are integers. Suppose that r is a real zero of f and that the leading coefficient of f is 1.Use the Rational Zeros Theorem to show that r is either an integer or an irrational number.
You can assume without loss of generality that has a non-zero constant term. In the case of a polynomial function that has a zero constant term, you can repeatedly factor out until you are left with a polynomial that does have a non-zero constant term or all of the possible values of are zero which is an integer.
The Rational Zeros Theorem says that if a polynomial function with integer coefficients has a lead coefficient of and a constant term coefficient of , and there exists a rational zero of that function, then the rational zero must have the form where is an integer factor of and is an integer factor of .
Since the leading coefficient of is given to be , if there is a rational zero for it must be an integer because in any possible zero , can only be equal to and since must be an integer, must be an integer.
If has no rational zeros or if is not an integer factor of , then must be irrational because and
John
My calculator said it, I believe it, that settles it
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