Not sure how to solve this one: If the remainder of dividing the polynomial $x^{2018}-ax^{2017}+bx^{2015}+c$ by $x^3-x^2+x-1$ is $ax+b$, where $a,b,c\in\mathbb{R}$, how much is $a-b+c$


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Firstly, notice that:

\begin{align*} x^{2018} - ax^{2017} + bx^{2015} + c = q(x)(x^{3} - x^{2} + x - 1) + ax + b \end{align*}

Secondly, notice that: \begin{align*} x^{3} - x^{2} + x - 1 = x^{2}(x-1) + (x - 1) = (x^{2} + 1)(x - 1) \end{align*}

  • $\begingroup$ Oh thank you, I got 3/2 as an answer, that seems right enough $\endgroup$ – Aleksa Oct 4 '18 at 22:58

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