Greatest common divisor of two polynomials

[Math Amateur]

I am working on Exercise 8 of Dummit and Foote Section 9.2 Exercise 8

====================================================================================

Determine the greatest common divisor of $$a(x) = x^3 - 2$$ and $$b(x) = x + 1$$ in $$\mathbb{Q} [x]$$

and write it as a linear combination (in $$\mathbb{Q} [x]$$ ) of a(x) and b(x).

=====================================================================================

In working on this I applied the Division Algorithm to a(x) and b(x) resulting in $$x^3 - 2 = (x^2 - x + 1) (x+ 1) + (-3)$$

then

$$(x + 1) = (1/3 x + 1/3) + 0$$Last non-zero remainder is -3

Therefore, gcd is -3

BUT!

This does not seem to be correct because -3 does not divide either a(x) and b(x)

Can someone please help?

Peter

 

[Opalg]

I am working on Exercise 8 of Dummit and Foote Section 9.2 Exercise 8

==================================================

Determine the greatest common divisor of $$a(x) = x^3 - 2$$ and $$b(x) = x + 1$$ in $$\mathbb{Q} [x]$$

and write it as a linear combination (in $$\mathbb{Q} [x]$$ ) of a(x) and b(x).

==================================================

In working on this I applied the Division Algorithm to a(x) and b(x) resulting in $$x^3 - 2 = (x^2 - x + 1) (x+ 1) + (-3)$$

then

$$(x + 1) = (1/3 x + 1/3) + 0$$Last non-zero remainder is -3

Therefore, gcd is -3

BUT!

This does not seem to be correct because -3 does not divide either a(x) and b(x)

Can someone please help?

Peter

$-3$ is a unit in $\mathbb{Q} [x]$, so is equivalent to $1$. You have shown that $$-\tfrac13(x^3-2) + \tfrac13(x^2-x+1)(x+1) = 1.$$ Thus $p(x)a(x)+q(x)b(x) = 1$, where $p(x) = -\frac13$ and $q(x) = \frac13(x^2-x+1)$. The polynomials $p(x)$ and $q(x)$ are both in $\mathbb{Q} [x]$.

 

[agentredlum]

I am an amateur but i tried doing your problem the way they do it here.

http://www.math.niu.edu/~beachy/abstract_algebra/study_guide/soln4.html

wolfram confirms their answer so it appears wolfram is giving answers over Q[x] for gcd command.

gcd (x^8-1, x^6 - 1) - Wolfram|Alpha

I am not encouraging you to cheat but WIA can be a valuable resource to check your work AFTER you get an answer by manual computation.

For your question WIA gives 1 as gcd.

gcd (x^3-2, x + 1) - Wolfram|Alpha

:)