Chapter 2.1 — Introduction to Polynomials

Testing Expressions Against the Rule

• Polynomials: 4x² + 5x − 2, y² − 8, the constant 5, 3x² + 5y, and 3xyz — every exponent present is a whole number.
• Not polynomials: 2x² + 3/x − 5 and 1/x + 1, since 3/x and 1/x rewrite as 3x⁻¹ and x⁻¹ — negative exponents.
• Not a polynomial: √x, since √x = x^(1/2) and ½ isn't a whole number.

Rewriting a term into exponent form before judging it is the reliable habit here — 3/x doesn't look like it has an exponent until it's rewritten as 3x⁻¹, at which point the negative power is obvious. The same rewriting trick works for roots: √x written as x^(1/2) immediately shows a fractional exponent, whereas x² and x³ obviously stay whole numbers no matter how they're rewritten. A quick two-question check handles almost every case that comes up: does any term have a variable in a denominator, and does any term have a variable under a root sign? A "yes" to either disqualifies the expression as a polynomial. It's the variable's exponent that matters, not the coefficient's — an expression like πx² + 5 is still a perfectly good polynomial even though π itself is irrational, because π is only multiplying x², not sitting as an exponent or underneath a root.

A Polynomial With Just One Variable

A variable is a symbol — typically x, y, or z — standing in for any real number. When only one such symbol appears throughout an expression, it's a polynomial in one variable: 3x² − 2x + 1, x⁴ − x² + 5x − 5, and x³ − x² are all examples.

The perimeter of a square, 4l, is a familiar instance of this shape — a fixed constant (4) multiplying a variable (l, the side length) that changes from square to square. When the constant itself isn't known in advance, letters like a, b, or c stand in for it instead, giving general one-variable expressions such as ax, by, or cz.

It's worth noticing what "one variable" does and doesn't mean here. 3x² − 2x + 1 has three terms but only one distinct symbol (x) running through all of them, so it's a one-variable polynomial with three terms — a trinomial. An expression like 2x² + 3xy + 5y² fails the one-variable test even though each individual term is otherwise a perfectly ordinary polynomial term, because x and y are two different symbols appearing side by side. Two polynomials with the same number of terms and the same degree can still differ completely in how many variables they involve — a distinction this chapter keeps separate from degree and term-count throughout.

Degree: One Term, Then the Whole Polynomial

The degree of a term is the sum of the exponents of every variable inside it — a term like −2xy³ has degree 1 + 3 = 4. The degree of a polynomial is simply the largest of these term-degrees across the whole expression.

Written in full generality, a one-variable polynomial of degree n looks like a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² + … + aₙ₋₁x + aₙ, where a₀ is never zero (otherwise the degree would really be lower than n). Two edge cases are worth remembering: a lone constant like −7 has degree 0, since −7 = −7x⁰, while the zero polynomial — the number 0 itself — has no defined degree at all, because 0 = 0×xⁿ for absolutely every value of n, so no single "largest" exponent exists to point to.

The standard-form expression is really just a compact way of saying "list every possible power of x from n down to 0, and allow any of those coefficients (except a₀) to be zero." A polynomial like 3x⁴ + 5x − 2 fits this form with a₁ = 0 and a₂ = 0 — the missing x³ and x² terms simply have coefficient zero, not "no term at all," which is exactly why the standard-form template can always list every power down to x⁰ without contradiction.

Naming Polynomials by Degree and by Term Count

Polynomials get two independent sets of names — one based on degree, one based on how many non-zero terms they have.

• By degree: constant (degree 0), linear (degree 1), quadratic (degree 2), cubic (degree 3).
• By term count: monomial (1 term), binomial (2 terms), trinomial (3 terms), multinomial (more than 3 terms).

The two labels are independent of each other — a binomial can be linear (x − 1, two terms, degree 1) or cubic (x³ + 2, two terms, degree 3) just as easily. A monomial of degree 10, such as 9x²y⁸ (exponents 2 and 8 sum to 10), and a trinomial of the same degree, such as 5 − x³ − 8x¹⁰, both satisfy "degree 10" while looking nothing alike — one term versus three, illustrating just how independent the two classification systems really are. One-variable polynomials are usually written in function notation — p(x), q(z), f(y) — purely as a naming convenience, so that a specific polynomial can be referred back to later without rewriting it out in full. Writing p(3), for instance, means "substitute 3 for x in whatever polynomial was called p(x)" — the notation is doing the same job as giving a variable a name, just one level up, for an entire expression rather than a single number.

One more distinction worth being precise about, since it trips learners up later: a binomial is defined purely by its term count (exactly two), never by its degree. πr² is a monomial (one term) despite involving the irrational constant π — the number of terms is what's being counted, not whether every part of the expression is a whole number itself. Likewise, "every polynomial is a binomial" is false in general, since a polynomial is free to have one term, two, three, or any larger number of terms; binomial is simply the name reserved for the two-term case.

Carrying These Definitions Into the Rest of the Chapter

Every remaining exercise in this chapter — finding a polynomial's zeroes, applying the Remainder and Factor Theorems, and factorising using identities — assumes the vocabulary built here: degree, term count, coefficient, and one-variable form. Exercise 2.1 drills exactly this vocabulary before the chapter moves on to zeroes. The same degree and classification ideas reappear, extended to a full theory of relationships between zeroes and coefficients, in Class 10's Polynomials chapter. Getting comfortable with degree and term-count now is what makes that later material readable on first pass, rather than requiring these same definitions to be relearned from scratch a year later.