What must be added to p(x)=x^4+2x^3-2x^2+x-1 to make it exactly divisible by g(x)=x^2+2x-3?
The book says that the remainder will be a linear polynomial and so the expression that must be added to p(x) will be of the form ax+b; the book essentially says we add the remainder. The divisor has roots 1 and -3. We can solve by putting p(1) and p(-3) equal to zero and finding the values of a and b. I know how to solve this but I am confused as to why we say that the expression to be added is of the form ax+b. How do we know this. Essentially what the book seems to be saying is add the remainder. But if we take the example7/3, the remainder is 1 but adding the remainder will not make 7 exactly divisible by 3. Here we subtract 1 from 3 and add the result to make 7 exactly divisible by 3. I believe the same principle must hold in the case of polynomials and the explanation in the book that we add the remainder to the dividend must be wrong. Please tell me the reason why we add ax+b. Please explain.
Also, can it be solved through polynomial long division? If so, how?
Are there any other easier ways to solve this? Feel free to ignore the last question and even the second last question if it will make the answer too long.