Exercise · ZW-EUCLIDEAN-DOMAIN-005 · ★☆☆ Basic
In \(\mathbb{Q}[x]\), what is the greatest common divisor of \(x^3-1\) and \(x^2-1\)?
Choose one option.
Show answer
- C. \(x - 1\)
Explanation
Dividing, \(x^3-1=x(x^2-1)+(x-1)\) and \(x^2-1=(x+1)(x-1)\), so the last nonzero remainder is \(x-1\). The polynomial \(x+1\) only divides \(x^2-1\).
Review the concept: Euclidean domain
An integral domain \(D\) with a norm \(N\) (a function from \(D\setminus\{0\}\) to the nonnegative integers) allowing division of \(a\) by \(b\neq0\) with quotient \(q\) and remainder \(r\) zero or of smaller norm than \(b\).
\[a=bq+r,\qquad r=0\ \text{ or }\ N(r)
More practice problems: euclidean domain
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