Adding & Subtracting Polynomials

Adding polynomials is just combining like terms.

(3x² + 5x − 2) + (x² − 3x + 7) = 4x² + 2x + 5.

Subtracting is the same *after* you distribute the minus sign to every term:

(3x² + 5x − 2) − (x² − 3x + 7)
= 3x² + 5x − 2 − x² + 3x − 7
= 2x² + 8x − 9.

Notice all three signs in the second bracket changed. Missing the second and third is the most common error in the whole topic. Write the distribution step out rather than doing it mentally.

Vocabulary: degree is the highest exponent, and a polynomial is usually written in descending order.

Polynomials form a ring under addition and multiplication: sums, differences and products of polynomials are polynomials, though quotients generally are not.

Addition is componentwise on coefficients of matching degree, which is why only like terms combine. Subtraction is addition of the additive inverse, and the inverse of a polynomial negates every coefficient — the formal statement of "distribute the minus."

Standard form arranges terms in descending degree. The degree of a sum is at most the larger of the two degrees, and can be smaller when leading terms cancel: (x² + 3x) + (−x² + 1) has degree 1.

Naming conventions used in later problems: by number of terms, monomial, binomial, trinomial; by degree, linear, quadratic, cubic, quartic. The leading coefficient is the coefficient of the highest-degree term and controls end behaviour, which becomes central when graphing polynomials in Algebra 2.