How do i form a polynomial equation given the details provided?

Question

My question: Write down a polynomial in $x$ which gives the remainder $3$ when it is divided by each of the following: $(x+1) (x-2) (x+3)$. What exactly does this mean? I know it wanted me to use the details provided to form an equation but how? From how i interpreted the questions, i have got $p(-1)=3, p(2)=3,p(-3)=3$. Is this the right first step? and how do I proceed with the next step? Please help! Thank you!

Answer 1

A polynomial $p(x)$ gives a reminder $r(x)$ upon dividing by $q(x)$ if there exists a polynomial $m(x)$ such that

$$p(x) = m(x)q(x) + r(x)$$

(sometimes you also want that $\deg r(x) \leq \deg q(x)$).

Anyway in your case there exist $m_1(x),m_2(x),m_3(x)$ such that

$$p(x) = m_1(x)(x+1)+3$$ $$p(x)= m_2(x)(x-2)+3$$ $$p(x) = m_3(x) (x+3)+3$$

In particular $(x+1)(x-2)(x+3)$ divides $p(x)-3$ this happens for example when

$$p(x) = (x+1)(x-2)(x+3)+3$$

Answer 2

How about:

$$p(x) = (x+1)(x-2)(x+3)+3$$