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## #1 2018-04-06 04:37:52

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

### help on proving this

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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## #2 2018-04-06 05:51:24

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

### Re: help on proving this

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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