Question about polynomial division and division theorem

Asked Jun 12 '22 at 00:55

Active Jun 12 '22 at 02:14

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For a certain polynomial, $P(3)=2$ and $P(-1)=-1$. Use this information to deduce the remainder when $P(x)$ is divided by $(x-3)(x+1)$.

I have tried to create a simultaneous equation with the division theorem, but I'm stuck. $$ P(x) = Q(x)(x-3) + 2 $$ and $$ P(x) = Q(x)(x+1) - 1 $$

edited Jun 12 '22 at 02:14

Sammy Black

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asked Jun 12 '22 at 00:55

user19282

What have you tried and where did you get stuck? – dxiv Jun 12 '22 at 01:01
I tried applying the division theorem and create a simultaneous equation, but I'm not sure on how to do that. – user19282 Jun 12 '22 at 01:21
2

Start by writing down the division of $P(x)$ by $(x-3)(x+1)$, then substitute $x=3$ and $x=-1$ and see what you get. Actually you should edit that into the question, otherwise it's going to get closed for lack of context and own work. – dxiv Jun 12 '22 at 01:27
If I substitute x=3 and x=-1, wouldn't that just give me 2 and -1, as stated in the question? Sorry if I'm missing something. – user19282 Jun 12 '22 at 01:30
Never mind, it equates the divisor and quotient to 0, and leaves the remainder. – user19282 Jun 12 '22 at 01:42
I still don't understand. I have P(3)=R(3) and P(-1)=R(-1)... – user19282 Jun 12 '22 at 01:43
You divide by a polynomial of degree $2$. What degree will the remainder $R(x)$ have? – dxiv Jun 12 '22 at 02:01
R(x) must have a lower degree than D(x), so it will either have a degree 0 or degree 1. – user19282 Jun 12 '22 at 02:04
Right, so $R(x)=ax+b$ and you know the values at two points, so you can determine $a,b$. – dxiv Jun 12 '22 at 02:06
So P(x)=Q(x)(x-3)(x-1)+ax+b. From here, do I substitute values? – user19282 Jun 12 '22 at 02:10
Please look at this post to learn how to typeset math on this site and elsewhere. – Sammy Black Jun 12 '22 at 02:15
You used $Q(x)$ for both quotients, but they are likely not meant to be the same polynomial. – Sammy Black Jun 12 '22 at 02:16
1

Thanks for the help everyone. I have solved it! – user19282 Jun 12 '22 at 02:23
See the linked dupe (and it many "Linked" posts) for the standard methods (and many worked examples). – Bill Dubuque Jun 12 '22 at 06:46

Question about polynomial division and division theorem

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