Exercise 261Algebra › PolynomialsHigh schoolAdded 3 Nov 2026

Degree of sums and products

#degree#operations#polynomial

Statement

Let $P(X)=3X^4-2X^2+X-1$ and $Q(X)=-X^3+4X^2+2$ .

Give the degree, leading coefficient, and constant term of $P$ and $Q$ .
Compute $\deg(P+Q)$ , $\deg(PQ)$ , $\deg(P-Q)$ .

Hint

For $P+Q$ , watch for cancellation of the leading terms.

Show solutionHide solution

Solution

1.
- $P$ : degree $4$ , leading coefficient $3$ , constant term $-1$ .
- $Q$ : degree $3$ , leading coefficient $-1$ , constant term $2$ .

2. Sum. $P+Q=3X^4-X^3+2X^2+X+1$ : degree $4$ (no cancellation in degree $4$ ).

Product. $\deg(PQ)=\deg P+\deg Q=4+3=7$ .

Difference. $P-Q=3X^4+X^3-6X^2+X-3$ : degree $4$ .

Remark. $\deg(P+Q)\le\max(\deg P,\deg Q)$ with strict inequality possible if the leading terms cancel. $\deg(PQ)=\deg P+\deg Q$ always (for polynomials over an integral domain).

Degree of sums and products

Statement

Solution

More problems in Polynomials