How does one determine the degree of a polynomial in a remainder theorem identity without using long division?
For example, a question asks:
Divide $2x^2 + 4x +5$ by $x^2-1$
Writing the remainder theorem identity, we get:
$2x^2 + 4x + 5 ≡ A (x+1)(x-1) + (Bx+C)$
I only knew that the identity was in the form, $2x^2 + 4x + 5 ≡ A (x+1)(x-1) + (Bx+C)$, after diving the polynomials together which gave me the answer, $2 + ((4x + 7)/ (x^2-1))$ and therefore knew the identity was in the form $A (x+1)(x-1) + (Bx+C)$
How would I be able to know what the identity is without dividing the polynomials using long division?
Best Answer
The simple answer is: you can't in general. You can know for sure that the remainder's degree will be at most $$\min\{\text{dividend's degree},\text{divisor's degree}-1\},$$ but it is possible that the remainder's degree will be less than that, or that there will be no remainder at all.