# Remainder theorem - Need Help

### Remainder theorem - Need Help

Need help on the below home work problem -
When P(x) is divided by x-1, the remainder is 1.
When divided by (x-2)(x-3), the remainder is 5.
What is the remainder when P(x) is divided by (x-1)(x-2)(x-5) ?
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### Re: Remainder theorem - Need Help

Good morning!

$$P(x)=q_1(x)(x-1)+1$$
$$P(x)=q_2(x)(x-2)(x-3)+5$$

$$P(x)=q_3(x)(x-1)(x-2)(x-3)+R(x)$$
$$R(x)=ax^2+bx+c$$

$$P(1)=1\\ P(2)=5\\ P(3)=5$$

$$\begin{cases}a+b+c=1\\4a+2b+c=5\\9a+3b+c=5\\\end{cases}$$

Solution:
$$\begin{cases}a=-2\\b=10\\c=-7\end{cases}$$

So:
$$R(x)=-2x^2+10x-7$$

Baltuilhe

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