1. ## Polynomial Remainder Theorem

Hi,

I hope someone can provide me with some clarification.

I'm trying to find the remainder for a dividend of x^2+9x-7. I already have information regarding the quotient and divisor: (x + 3) and (x + 5). It's unknown which one is the quotient or divisor, but I don't think that's relevant information anyways. One method of solving the remainder would be to isolate the remainder ("r") in the given equation: (x + 3)(x + 5) + r = x^2 + 9x - 7. This works for me, and I get the correct answer of r = x - 22.

However, I don't understand why I ALSO can't use the remainder theorem to solve this. While I understand that a solution with a variable is impossible to solve with the remainder theorem, how would I know this prior to using the theorem? I thought that when factors have a degree of 1, that you could find the remainder by simply using the remainder theorem - clearly this case is different. Please help!!!!

Sincerely,
Olivia

2. ## Re: Polynomial Remainder Theorem

performing synthetic division with x = -3 ...

Code:
-3].......1........9........-7
..................-3.......-18
-------------------------------
...........1.......6.......-25
note the remainder, -25 = x - 22

performing synthetic division with x = -5 ...

Code:
-5].......1........9........-7
..................-5.......-20
-------------------------------
...........1.......4.......-27
note the remainder, -27 = x - 22

3. ## Re: Polynomial Remainder Theorem

It seems like an odd question because the remainder should be of a lower order than the divisor.

If $\displaystyle (x+3)$ is the divisor, then your remainder $\displaystyle x-22 = (x+3) - 25$. So the quotient ought to be $\displaystyle (x+6)$ rather than $\displaystyle (x+5)$ giving a remainder of $\displaystyle -25$.

Similarly, if the divisor is $\displaystyle (x+5)$, then the remainder $\displaystyle x-22 = (x+5) - 27$. So the quotient ought to be $\displaystyle (x+4)$ rather than $\displaystyle (x+3)$ giving a remainder of $\displaystyle -27$.

4. ## Re: Polynomial Remainder Theorem

Yes, the fact that the remainder isn't lower than divisor is what makes it confusing. So are you suggesting that the question is not explained correctly? (i.e. I need to change quotient in order for me to produce x - 22 ?)

Also, how would I know in advance that I can't use the remainder theorem for this problem?