Polynomial remainder problem

Hello, this is a problem from Israel Gelfand's Algebra book about polynomials.

Problem 152. The polynomial gives a remainder of when divided by . Find the remainder when is divided by .

My try: We know that where Q is the quotient, this is the same as , I see that all I have to do is to rewrite this equation to something like this , I tried dividing both sides by but I can't get the form of .

Thanks in advance.

Re: Polynomial remainder problem

So the remainder must be -2.

Re: Polynomial remainder problem

Quote:

Originally Posted by

**Prove It**
So the remainder must be -12.

Thank you for the reply Prove It, but I have few questions.

, isn't this correct? instead of .

Re: Polynomial remainder problem

Quote:

Originally Posted by

**PaulAdrienMauriceDirac** Thank you for the reply Prove It, but I have few questions.

, isn't this correct? instead of

.

Oops that's a typo. I'll fix it.

Re: Polynomial remainder problem

Re: Polynomial remainder problem

Quote:

Originally Posted by

**PaulAdrienMauriceDirac** It's ok, so should it be something like this?

and the remainder is -2? I don't understand this, can you explain me why is remainder -2 and not

? Is it because we divided

by

in the left side?

The remainder is the numerator of the NON-WHOLE part. 5 is whole.

Re: Polynomial remainder problem

Quote:

Originally Posted by

**Prove It** The remainder is the numerator of the NON-WHOLE part. 5 is whole.

Hm, can you tell me how did you know that, is that a rule in this type of situations, that remainder is the numerator of the non-whole part?

Re: Polynomial remainder problem

Quote:

Originally Posted by

**PaulAdrienMauriceDirac** Hm, can you tell me how did you know that, is that a rule in this type of situations, that remainder is the numerator of the non-whole part?

It's a basic rule of writing improper fractions as mixed numbers.

Re: Polynomial remainder problem

Quote:

Originally Posted by

**Prove It** It's a basic rule of writing improper fractions as mixed numbers.

Yes I know that rule, I was just confused since I learned that , I was confused about which one was quotient and which one was remainder, now I understand, thank you!

Re: Polynomial remainder problem

Remember that remainder is always less than the divisor, this is a little difficult to formalize in case of polynomials rather than in case of numbers. I think, a remainder when divided with a linear polynomial would be a constant, the acceptable remainder when you divide using a quadratic (like x^2 - 1) would be a linear polynomial (5x - 7) or a constant. Keep this in mind, and you will be able to factorize out correctly.

Salahuddin

Maths online