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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?

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  • $\begingroup$ The degree of the remainder should be $=$ min(the degree of the divisor $-1,$ the degree of the dividend) $\endgroup$ – lab bhattacharjee Jul 28 '13 at 13:49
  • $\begingroup$ Not so, @lab. Consider $$\frac{x^2+2}{x^2+1}=1+\frac{1}{x^2+1}$$ $\endgroup$ – Cameron Buie Jul 28 '13 at 14:04
  • $\begingroup$ Good point @CameronBuie. Do you know how to determine the identity without dividing the polynomials using long division? $\endgroup$ – PurpleJess Jul 28 '13 at 15:46
  • $\begingroup$ There is no generally applicable method of doing so. See my answer below. $\endgroup$ – Cameron Buie Jul 29 '13 at 0:37
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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.

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