Question

Express f as gq+r in the following cases:
i) f = x4+x+1, g = x2+1 in Q[x].
ii) f = 5x3+2x2+3x+1, g = x+2 in Z7[x].

Answer 1

Answer on Question #44362 - Math, Abstract Algebra

Problem

Express $f$ as $gq+r$ in the following cases:

i) $f = x4+x+1$, $g = x2+1$ in $\mathbb{Q}[x]$.

ii) $f = 5x3 + 2x2 + 3x + 1$, $g = x + 2$ in $Z7[x]$.

Solution

i) $x^4 + x + 1 = x^4 - 1 + x + 2 = (x^2 + 1)(x^2 - 1) + (x + 2)$;

ii) $5x^{3} + 2x^{2} + 3x + 1 \equiv (x + 2)(5x^{2} + 6x + 5) + 5$ in $\mathbb{Z}_7[x]$.

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