= Degree of a polynomial
{title2=$\deg$}
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= Degree
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= Polynomial degree
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The degree of a nonzero <polynomial> is the largest exponent of a <monomial> with a nonzero <coefficient>. Thus $X^3-3X^2+4$ has degree three, while any nonzero constant <polynomial> has degree zero. The zero <polynomial> is treated separately in degree statements.
Over an <integral domain>, <polynomial degrees> add under <multiplication>: $\deg(pq)=\deg p+\deg q$ for nonzero factors, because the product of their <leading coefficients> cannot vanish. Under <addition>, cancellation may lower the degree: $(X^2+1)+(-X^2+X)=X+1$. This explains why degree is reliable for counting factors and bounding <polynomial roots>, but cannot always be inferred from the degrees of summands alone.
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