Finding the GCD of two polynomials.

Question

Hello I'm trying to find the gcd of these two polynomials:

$$x^4-x^3-4x^2-x+5$$ $$x^2+x-2$$

And then I want to express the gcd of these two polynomials in terms of themselves multiplied by other polynomials (Like Bezout's lemma.)

I am struggling to do this though and would like so help. Many thanks.

Answer 1

Let $f(x)=x^4-x^3-4x^2-x+5$, $g(x)=x^2+x-2$

For $g(x)$, as $g(1)=0$,

$g(x)=(x-1)(x+2)$

For $f(x)$, observe

$f(1)=1-1-4-1+5=0$, so $(x-1)$ is a factor of $f(x)$, by the Remainder Theorem.

Hence $f(x)=(x-1)(x^3-4x-5)$.

For $h(x)=x^3-4x-5$,

Since $h(2)=8-8-5=-5\neq0$

$x-2$ obviously cannot divide $x^3-4x-5$ and they have no common factor.

So the GCD of $g(x)$ and $f(x)$ is $x-1$

Answer 2

Use the Division algorithm to divide the first by the second and keep going until we get zero remainder. The last we divided by is the gcd. Then we go back step by step. to express gcd as a linear combination.