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I need to find an equation, given the information that when divided by (x-1) it has a remainder of 2, and when divided by (x+2) the remainder is -19. I've looked at many answers to this type of question but none of them have really explained what to do. Please help!

Gilsin
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  • Can you link some of the answers you've looked at? What were your thoughts about them? – uriyabsc Dec 08 '20 at 20:01
  • This question seems most similar: https://math.stackexchange.com/questions/1103023/find-a-polynomial-given-the-remainders-of-the-division-of-that-polynomial-with-3, however there are 3 dividends instead of 2. The op also didn't say how they found the degree of the equation, and those who commented didn't show how to get a final answer. – Gilsin Dec 08 '20 at 20:09
  • The degree of the polynomial is not determined from the given data. If you have one solution, then adding any (polynomial) multiple of $(x-1)(x+2)$ will give another. But if all you want is to find one solution, start by writing $p(x)=q(x)(x-1)+2$. and then note that we also have $p(-2)=-19$. – lulu Dec 08 '20 at 20:11
  • I'm still not sure where to go from here since I don't have a quotient. – Gilsin Dec 08 '20 at 20:17
  • the quotient is not determined, but it doesn't have to be. Just write out what $p(-2)=-19$ means using the expression with quotient and remainder. That will tell you what $q(-2)$ is. Then, any $q(x)$ with the right value at $-2$ will work. – lulu Dec 08 '20 at 20:22
  • Note: Worth remarking that, since only two values were provided, there is a linear polynomial $p(n)=an+b$ that works. That corresponds to the case $q(x)$ constant, in my set up, But you can find the linear solution without all that fuss. – lulu Dec 08 '20 at 20:26

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Let $n$ and $m$ be two natural numbers which respectively represent the dividends of the polynomial we're looking for, we can write two expressions for that polynomial:

$$p_1= n(x-1) + 2$$ $$p_2= m(x+2) - 19$$

Simplifying further yields:

$$p_1= nx + (2-n)$$ $$p_2= mx+(2m-19)$$

And since, by our definition, $p_1=p_2$:

$$ \left\{ \begin{array}{c} n=m \\ 2-n=2m-19 \end{array} \right. $$

The above is true, because 2 polynomials are equal to each other only if their corresponding coefficients are the same.

Solving the equation system above yields $n=m=7$ (2 natural numbers!), and therefore, the polynomial that satisfies the answer for this question would be:

$$p= p1 = 7(x-1)+2 = 7x-5$$

uriyabsc
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