1.43 Polynomial Addition and Subtraction Definitions

Polynomial Addition and Subtraction Definitions establishes the vocabulary for combining two polynomials into a single resulting polynomial through addition or subtraction, and describes the specific transformation applied to a polynomial's terms that makes subtraction possible using the same combining process as addition.

Polynomial Sum

A polynomial sum is the result of adding two polynomials together, obtained by combining every like term that appears between the two polynomials while keeping any unlike terms unchanged in the result. Adding (3x² + 2x − 1) and (x² − 5x + 4) combines the x² terms, the x terms, and the constant terms separately, producing the polynomial sum 4x² − 3x + 3.

Polynomial Difference

A polynomial difference is the result of subtracting one polynomial from another, obtained by combining every like term between the two polynomials according to the subtraction, again leaving any unlike terms unchanged in the result. Subtracting (x² − 5x + 4) from (3x² + 2x − 1) requires subtracting each corresponding like term individually, producing the polynomial difference 2x² + 7x − 5.

Polynomial Additive Inverse

The polynomial additive inverse of a given polynomial is the polynomial obtained by reversing the sign of every one of its terms, such that adding the original polynomial and its additive inverse together produces the zero polynomial. The additive inverse of x² − 5x + 4 is −x² + 5x − 4, and taking advantage of this additive inverse allows polynomial subtraction to be carried out as an addition: subtracting a polynomial is equivalent to adding its additive inverse.

Together, these definitions describe polynomial addition and subtraction as processes of combining like terms across two polynomials, with the polynomial additive inverse establishing that subtraction is simply addition performed against a sign-reversed version of the second polynomial, unifying both operations under a single combining procedure.