Factoring a Polynomial with Non-Integer Roots

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Homework Help Overview

The discussion revolves around factoring a polynomial, specifically f(t) = t^3 - 6t^2 + 9t + 2, which is noted to have non-integer roots. Participants are exploring the nature of the roots and the factorability of the polynomial.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants are examining potential rational roots using the Rational Root Theorem and discussing the implications of finding no rational roots. There is also a mention of the polynomial being factorable over some field, prompting questions about the nature of its roots.

Discussion Status

The discussion is active with differing opinions on the factorability of the polynomial. Some participants assert that it is not factorable with integer or rational roots, while others suggest that it is factorable in a broader context, indicating a productive exploration of the topic.

Contextual Notes

Participants are working under the assumption that the polynomial has no rational roots based on initial evaluations, which may influence their approaches to factoring it. There is also an acknowledgment of the polynomial's complexity due to its non-integer roots.

Chocolaty
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f(t) = t^3 - 6t^2 +9t + 2

f(1) is not equal to 0
f(-1) is not equal to 0
f(2) is not equal to 0
f(-2) is not equal to 0
No common factor, can't group either

How can I work this one out?
thx
 
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Here's something to remember.
If f=a_0+a_1x+...+a_n x^n is a polynomial with integer coefficients of degree >=1 and q=b/c (ggd(b,c)=1) is a rational root, then b|a_0 and c|a_n.
So the possible rational roots of your polynomial are [itex]\pm 2[/itex], which not roots. This means there are no rational roots so it's a nasty polynomial.
 
I don't believe this is factorable
 
Of course it's factorable, over some field, it's just that the roots aren't integers (or rational). There's even a formula for the roots. try googling for it.
 

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