The highest nonzero exponent is the degree of the polynomial.
As a convention, the degree of a zero polynomial is said to be -∞.
Lemma 1: deg(A+B) ≤ max(degA,degB)
Proof:
(1) We start with with deg A = 0, deg B=0 where A ≠ -B [We can ignore deg = - ∞ since by the Additive Identity Property (see here), it is true]
(2) A+B ≠ 0, so deg(A+B) = 0
(3) Assume that this is true up to deg A, deg B ≤ n-1.
(4) Let:
A = anxn + ... + a0
B = bnxn-1 + ... + b0
(5) We can assume that an ≠ -bn [Since if this is the case, the n degrees cancels out and the proposition is true by the inductive hypothesis.]
(6) an + bn = (a+b)n [See here for Additive Rule for Polynomials]
(7) So the deg(A+B) = n
QED
Lemma 2: deg(AB) = deg(A) + deg(B)
Proof:
(1) Assume deg A = 0, deg B = 0
(2) Then deg(AB) = 0 = 0 + 0
(3) Assume that the proposition is true up to n-1
(4) Let:
A = anxn + ... + a0
B = bnxn-1 + ... + b0
(5) The highest exponent will be an*bn = (a*b)n+n
(6) deg(a*b)n+n = 2n = deg(a) + deg(b)
QED
References
- "Degree of a Polynomial", Wikipedia.org
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