A sum of simple pieces
A polynomial is built from one ingredient repeated: a number times a whole-number power of a variable. Each such piece — like 5, or 3x, or −7x^2 — is a term. String a few of them together with plus and minus signs and you have a polynomial: 3x^2 − 7x + 5. The expression 1/x or sqrt(x) is not a polynomial, because the exponent on x must be a whole number (0, 1, 2, 3, …) — never negative, never a fraction.
The number multiplying a term is its coefficient. In −7x the coefficient is −7; the sign always travels with the number. A term with no visible variable, like the 5, is the constant term — you can think of it as 5x^0, since x^0 = 1.
Counting terms, measuring degree
We name short polynomials by how many terms they have. One term is a monomial (4x^3). Two terms is a binomial (x − 9). Three terms is a trinomial (x^2 + 5x − 6). Past three we usually just say “a polynomial.”
The degree is the largest exponent that appears. In 3x^2 − 7x + 5 the degree is 2, because x^2 is the highest power. The coefficient sitting on that highest-degree term is the leading coefficient — here it is 3. Degree tells you a lot at a glance: degree 1 is a straight line, degree 2 a parabola, degree 3 a cubic.
Polynomial: 3x^2 − 7x + 5 terms: 3x^2 , −7x , +5 coefficients: 3 , −7 , 5 degree: 2 1 0 → highest is 2 leading coeff: 3 (sits on x^2) constant term: 5 name: trinomial (3 terms)
Standard form
Standard form means writing the terms in descending order of exponent — highest power first, constant last. The messy 5 − 7x + 3x^2 becomes the tidy 3x^2 − 7x + 5. Same polynomial, just combed into order so the degree and leading coefficient are obvious.
- Find the exponent on each term (a bare number is exponent 0).
- Reorder the terms from the largest exponent down to the smallest.
- Keep each term’s own sign attached as you move it.
- Read off the degree (first exponent) and the leading coefficient (first number).