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**G123****Member**- Registered: 2018-04-06
- Posts: 2

Let f(x) be a quartic polynomial with integer coefficients and four integer roots. Suppose the constant term of f(x) is 6.

(a) Is it possible for x=3 to be a root of f(x)?

(b) Is it possible for x=3 to be a double root of f(x)? Prove your answers.

Is there a way to prove this question with the Rational root theorem?

Edit 2: I solved (a) but I'm not sure how to solve part (b)

*Last edited by G123 (2018-04-06 05:47:53)*

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,462

hi G123

Welcome to the forum.

Yes, you can use the rational root theorem. You'll find it here:

https://en.wikipedia.org/wiki/Rational_root_theorem

If that leaves your head spinning let's simplify by looking at a quartic equation.

If p/q is a rational solution, ie p and q are integers

and if we times by q^4

If we assume p/q is in its lowest terms then => q divides a and p divides e.

But you are told that the solutions are all integers; as all integers are rationals this means that q = 1 and p divides e.

So could p be 3 ? Yes as 3 divides 6

Could p be 9 (ie a repeated root of 3) ? No because 9 doesn't divide 6.

Hope that helps,

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob Bundy

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