Chapter 3.1 — Introduction to Polynomials

What Counts as a Polynomial

A polynomial in x is a sum of finitely many terms of the form axⁿ, where a is a real number and n is a whole number — 0, 1, 2, 3, and so on, never negative and never a fraction. 3x² − 1, 7x − 1/2, and x³ − x² + x − 1 all qualify. 3/x² − 1, x²⁰ − 5x⁻³, and 1/x − 1 don't, because each contains a negative power of x. Notice that the coefficient (7x − 1/2's "7") can be any real number at all, including a fraction — it's specifically the exponent that must be a whole number, never the coefficient sitting in front of the variable.

• 2x³ — a polynomial; the power 3 is a whole number.
• 1/(x − 1) — not a polynomial; this is 1 divided by an expression involving x, equivalent to a negative power.
• 4z² + 1/7 — a polynomial; 1/7 is just a constant, not a power of z.
• m² − √2m + 2 — a polynomial; √2 is simply a real-number coefficient, and coefficients are allowed to be irrational.
• p⁻² + 1 — not a polynomial; the exponent −2 is negative.

Notice that the disqualifying feature is always about the exponent, never about how complicated an expression looks. 4z² + 1/7 might look busier than p⁻² + 1 at first glance, but it's the perfectly ordinary one — the fraction sits on a plain number, not on a power of the variable. Checking every single exponent in an expression, not just the largest or most obvious one, is the only reliable way to classify a borderline case correctly.

Naming a Polynomial by Its Degree

The degree of a polynomial is the highest power of x that appears in it, and the degree gives the polynomial its name.

The condition a₀ ≠ 0 matters more than it looks: if the leading coefficient were allowed to be zero, the highest-power term would simply vanish, and the polynomial's real degree would be something lower than n — which defeats the point of calling it "degree n" in the first place.

Degree and number of terms are easy to conflate but measure completely different things. 7z⁴ − 3z² has only two terms yet degree 4, because degree only ever looks at the single highest power present — the count of terms written down is irrelevant to it. A polynomial with a hundred terms could still have degree 2, provided none of those terms involves a power higher than x². This distinction resurfaces directly as a true/false trap in the exercises that follow, so it's worth internalising now rather than relearning it under pressure later.

Evaluating a Polynomial at a Point

If p(x) is a polynomial and k is a real number, replacing every x with k throughout the expression and simplifying gives the value of p(x) at x = k, written p(k), a piece of notation used constantly from here onward.

• p(x) = x² − 2 → p(2) = 2² − 2 = 4 − 2 = 2
• f(y) = y³ + 8y + 1 → f(3) = 27 + 24 + 1 = 52
• g(m) = m⁴ − 1 → g(1) = 1 − 1 = 0

That last result — g(1) = 0 — isn't just an ordinary value. Whenever substituting a number into a polynomial produces exactly zero, that number earns a special name of its own, covered next. Notice too how each example uses a different letter for its variable — x, y, m — purely as a labelling choice; the rules for evaluating, naming the degree, and finding zeroes are exactly the same regardless of which letter is used.

A Full Table of Values

Evaluating the same polynomial at several points side by side shows how its output rises and falls. For p(x) = x² − 5x − 6:

Reading across, the values fall from 18 down to −12 and start levelling off — but the single most important entry in this table is p(−1) = 0, which leads directly into the next idea. Notice also that p(2) and p(3) both equal −12, the same output from two different inputs — a completely ordinary occurrence for a polynomial, since nothing in the definition requires every output value to be unique. Only when the output is specifically zero does the input value earn special attention.

Zeroes: Where a Polynomial Vanishes

A real number k is a zero of p(x) if p(k) = 0. From the table above, x = −1 is a zero of x² − 5x − 6, since substituting it gives exactly 0.

• p(x) = x² − 4 → p(2) = 0 and p(−2) = 0, so 2 and −2 are both zeroes.
• p(x) = x² − 4x + 3 → p(1) = 0 and p(3) = 0, so 1 and 3 are its zeroes.
• p(x) = x² − 9 → checking x = 3 and x = −3 directly: both give 9 − 9 = 0, confirming both are zeroes.

Every example above is a quadratic with exactly two zeroes — no coincidence, since a quadratic polynomial can never have more than two. In general, a polynomial's degree sets a hard ceiling on how many zeroes it can have: a linear polynomial has at most one, a quadratic at most two, a cubic at most three, and so on. Some polynomials fall short of that ceiling — a quadratic can have one repeated zero, or even none at all among the real numbers — but no polynomial ever exceeds it.

Carrying This Forward

This vocabulary — degree, value, zero — is used without re-explanation throughout the rest of the chapter, so it's worth being genuinely comfortable with all three before moving on, rather than treating this introduction as background reading to skim past. Exercise 3.1 drills identifying coefficients and degree, then verifying zeroes by direct substitution; Exercise 3.2 connects zeroes to where a polynomial's graph crosses the x-axis; Exercise 3.3 uncovers a direct formula linking zeroes to coefficients, skipping substitution entirely; and Exercise 3.4 closes the chapter with polynomial long division. If you'd like a refresher on the equivalent ideas from Class 9, see Class 9's Polynomials introduction, which covers degree and zeroes at a gentler pace, using smaller and more familiar numbers, before this chapter builds on them directly.