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

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.

Answer by Solver92311(821) (Show Source): You can put this solution on YOUR website!

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

From
I > Ø

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