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Basically, I got this from a olympiad practice test and I still don't understand the logic of it, so the question asks us to find the remainder of the following:

$$\frac{x^{74} + 23}{x + 1}$$

At first I thought I could just use the big brain shortcut of replacing x with 1, but the remainder turned out to be $0$, and the multiple choice options only has the answers $74$, $25$, $24$, and $23$. If you're going to use something like the modular operation please explain it to me because I'm not familiar to it.

Any help would be extrememly appreciated :)

AlexJaynMF
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2 Answers2

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When $f(x)$ is divided by $x-a$, the remainder is $f(a)$. Here, $f(x)=x^{74}+23$ so the remainder when it is divided by $(x+1)$ is $f(-1)=24.$ This is known as remainder theorem.

Z Ahmed
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$\frac{p(x)}{x+1}=c(x)+\frac{r(x)}{x+1}\iff p(x)=c(x)(x+1)+r(x)$ where $\text{deg}(r(x))<\text{deg}(x+1)=1$, so $\text{deg}(r)=0$, $r(x)=r\in\Bbb R$. So $r=p(x)-c(x)(x+1)$ for all $x$ and, in particular, $r=p(-1)-c(-1)(-1+1)=p(-1)$.

This is essentially the proof of the Remainder Theorem.

Tito Eliatron
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