Polynomials: terms and degree

Powers of x, scaled and added — that's a polynomial. 3x² − 5x + 7 is one; so is x¹⁰⁰, and so is the humble 4. They're the expressions algebra can fully handle with just +, −, and ×, which makes them the material everything in this module is built from.

The parts

Take 3x² − 5x + 7:

Degree names the family: degree 1 is linear (a line), degree 2 quadratic (a parabola), degree 3 cubic. The degree is the polynomial's single most telling number — it caps how many times the graph can turn and how many roots the polynomial can have.

Like terms: the only things that combine

3x² and 5x² are like terms — same variable, same power — so they add: 8x². But 3x² + 5x stays as it is. x² and x are different objects, as uncombinable as meters and square meters (that's literally what they are, in the area model coming two lessons from now).

Simplifying (2x² + 3x − 1) + (x² − 5x + 4):

$2x^2 + 3x - 1 + x^2 - 5x + 4 = 3x^2 - 2x + 3$

Subtracting works the same after one care step: distribute the minus onto every term first. (2x² + 3x) − (x² − 4x) = 2x² + 3x − x² + 4x = x² + 7x. The forgotten sign-flip on the second term is the classic error.

Standard form

Write terms from highest power to lowest: x³ − 2x² + x − 5. Nothing about the math requires it, but everyone reads polynomials this way — degree visible at the front, constant at the back — and comparing or adding them becomes column work, like lining up place values.

Why this matters

Polynomials are the nouns of the next three modules: you'll multiply them, factor them, and solve them. Terms, coefficients, and degree is the vocabulary every one of those skills is described in.