I have the next problem. Determine the gcd $d(x)$ of two polynomials with real coefficients $a(x) = x^{4}-1$ and $b(x) = x^{3}-x^{2}-x+1$. And then, determine two polynomials with real coefficients $\alpha(x)$ and $\beta(x)$ to describe $d(x)$ as a combination of $a(x)$ and $b(x)$

I calculated the gcd of $a(x)$ and $b(x)$ using the euclidean algorithm and I have that $d(x) = 2x^{2}-2$. But now, I don't know how to proceed. I need to calculate $$\alpha(x)\cdot a(x)+\beta(x)\cdot b(x) = d(x) $$

or not? If anyone can give me an explanation I would really appreciate it :)


I have the answer. I can write $$a(x) = b(x)(x+1)+d(x) $$

So $$d(x) = a(x)-b(x)(x+1) $$

Then, $\alpha(x) = 1$ and $\beta(x)=-(x+1)$. It's really easy :)


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