Abstract Algebra: Theory and Applications · Problem 17.5.29

Integral Domains — Thomas Judson

Mathematics — Abstract Algebra · polynomial-degree

Problem

Let p(x)p(x) and q(x)q(x) be polynomials in R[x]R[x], where RR is a commutative ring with identity. Prove that deg⁡(p(x)+q(x))≤max⁡(deg⁡p(x),deg⁡q(x))\deg(p(x) + q(x)) \leq \max(\deg p(x), \deg q(x)).
Intuitive explanation

The key insight is that polynomial degree is determined by the leading term—the highest power with a nonzero coefficient. When you add two polynomials, the leading terms either cancel (if they have the same degree and opposite coefficients) or one dominates the sum. Either way, no new high-degree terms appear, so the sum's degree can't exceed the larger of the two original degrees.

Intuitive explanation

The key insight is that polynomial degree is determined by the leading term—the highest power with a nonzero coefficient. When you add two polynomials, the leading terms either cancel (if they have the same degree and opposite coefficients) or one dominates the sum. Either way, no new high-degree terms appear, so the sum's degree can't exceed the larger of the two original degrees.

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