A 9th-century Baghdad scholar wrote the first systematic treatment of polynomials — and gave us the word "algebra" along the way.
The types of polynomials fall into two independent classification systems. Degree tells you the highest power of the variable: degree 0 (constant), degree 1 (linear), degree 2 (quadratic), degree 3 (cubic), degree 4 (quartic), degree 5 (quintic), and higher. Number of terms tells you how many separate algebraic chunks the polynomial contains: 1 (monomial), 2 (binomial), 3 (trinomial), 4 or more (just called a polynomial).
The two classifications are independent. A polynomial is both a "cubic" AND a "trinomial" simultaneously — those categories describe different features. The full label often combines both, like "quadratic binomial" for $x^2 - 9$.
Classification by Degree
The degree of a polynomial is the highest exponent of the variable. Standard names by degree:
Degree | Name | Example |
|---|---|---|
0 | Constant | $7$ |
1 | Linear | $3x + 4$ |
2 | Quadratic | $x^2 - 5x + 6$ |
3 | Cubic | $2x^3 + x - 1$ |
4 | Quartic (or biquadratic) | $x^4 - 3x^2 + 2$ |
5 | Quintic | $x^5 + 2x^4 - x$ |
6 | Sextic | $x^6 - 1$ |
7 | Septic | $x^7 + x$ |
$n$ | Degree-$n$ polynomial | $x^n + \cdots$ |
The zero polynomial ($p(x) = 0$) has no degree — or, depending on convention, is assigned degree $-\infty$. It is the only polynomial without a leading term.
Special degree categories:
Constant polynomial — degree 0. Just a number with no variable, like $5$ or $-\frac{3}{2}$.
Zero polynomial — the polynomial that is identically zero. Distinct from a constant polynomial.
Classification by Number of Terms
A term in a polynomial is a single algebraic chunk separated by a $+$ or $-$ sign. Counting terms gives the second classification:
Number of terms | Name | Example |
|---|---|---|
1 | Monomial | $5x^3$ |
2 | Binomial | $x^2 - 9$ |
3 | Trinomial | $x^2 + 5x + 6$ |
4+ | Polynomial (no special name) | $x^3 + 2x^2 - 7x + 4$ |
Note that the term-count classification only applies after like terms are combined. $3x^2 + 5x^2 + 7$ has three written terms but is really a binomial — once you combine, it becomes $8x^2 + 7$, two terms.
The Two-Word Combined Label
Every polynomial carries both labels simultaneously. Some examples:
$7$ — constant monomial (degree 0, 1 term)
$3x$ — linear monomial
$x + 2$ — linear binomial
$x^2 - 9$ — quadratic binomial
$x^2 - 5x + 6$ — quadratic trinomial
$2x^3$ — cubic monomial
$x^3 + 1$ — cubic binomial
$x^3 - 6x^2 + 11x - 6$ — cubic polynomial (4 terms, no special name)
From Easy to Tricky — Examples
Quick. Classify $7x^2 - 3x + 2$ by degree and by number of terms.
The highest power of $x$ is 2 — so it is a quadratic (degree 2). It has three terms — so it is a trinomial.
Final answer: quadratic trinomial.
Standard (Wrong-Path-First). Classify $4x^3 + 2x^2 - x^2 + 5$.
Wrong path: the rusher counts written terms: $4x^3$, $2x^2$, $-x^2$, $5$. Four terms. Degree 3. Calls it a "cubic polynomial" (4 terms = no special name).
The slip: like terms $2x^2$ and $-x^2$ haven't been combined. The polynomial is not yet in simplified form, so the count is wrong.
Correct path. Combine like terms first:
$4x^3 + 2x^2 - x^2 + 5 = 4x^3 + x^2 + 5$
Now count: $4x^3$, $x^2$, $5$ — three terms. Degree 3.
Final answer: cubic trinomial.
In our Grade 8 cohort, this is the single most common slip on classification questions — students classify before simplifying. The fix is a one-line rule: combine first, classify second. The memorizer who memorised "count the terms" applies the rule strictly without the prerequisite simplification.
Stretch. A polynomial $p(x)$ satisfies: when added to $3x^2 - 2x + 1$, the result is $5x^3 + x^2 + 4x - 7$. Find $p(x)$ and classify it.
Set up: $p(x) + (3x^2 - 2x + 1) = 5x^3 + x^2 + 4x - 7$.
Solve for $p(x)$ by subtracting:
$p(x) = (5x^3 + x^2 + 4x - 7) - (3x^2 - 2x + 1)$
$= 5x^3 + x^2 - 3x^2 + 4x + 2x - 7 - 1$
$= 5x^3 - 2x^2 + 6x - 8$
Classify: degree 3, four terms.
Final answer: $p(x) = 5x^3 - 2x^2 + 6x - 8$ — a cubic polynomial (no special term-count name).
Why Classifying Polynomials Matters
"Different degrees, different toolkits."
The point of classifying polynomials isn't pedagogy for its own sake — it tells you which solution method to reach for.
Linear polynomials are solved by isolating the variable — one step, always works.
Quadratic polynomials have the quadratic formula, factoring, or completing the square — all reliable.
Cubic and quartic polynomials have closed-form solutions (Cardano's formula for cubics, Ferrari's for quartics) but are usually factored using the Rational Root Theorem.
Quintic and higher polynomials have no general closed-form solution — Évariste Galois proved this in 1832. They must be solved numerically.
The degree determines which method works. The classification by terms tells you which factoring shortcuts apply — sum and difference of squares for quadratic binomials, sum and difference of cubes for cubic binomials, and so on.
The NIST Digital Library of Mathematical Functions tabulates polynomial families precisely because the type determines the algorithm. Recognise the type, save yourself an hour of wrong-method attempts.
The Slip-Ups That Cost Marks on Classifying Polynomials
Mistake 1: Classifying before combining like terms.
Where it slips in: any polynomial where like terms appear separately in the original expression.
Don't do this: count $4x^3 + 2x^2 - x^2 + 5$ as four terms.
The correct way: simplify first. After combining, the polynomial is $4x^3 + x^2 + 5$ — three terms, classified as a cubic trinomial.
Mistake 2: Treating degree and term-count as competing categories.
Where it slips in: students who think "cubic" and "trinomial" can't both describe the same polynomial.
Don't do this: ask "is $x^3 - 5x^2 + 6x$ a cubic or a trinomial?"
The correct way: the answer is both. Degree (cubic) and number of terms (trinomial) are independent classifications. The second-guesser will sometimes try to pick one label and ignore the other — list both labels in the answer.
Mistake 3: Calling a non-polynomial a "polynomial".
Where it slips in: expressions with negative or fractional exponents on the variable, like $x^{-2}$ or $\sqrt{x}$.
Don't do this: classify $x^{-2} + 3x$ as "a polynomial with negative-degree term".
The correct way: a polynomial requires non-negative integer exponents on all variables. $x^{-2}$ has exponent $-2$ — that is not a polynomial. $\sqrt{x} = x^{1/2}$ has a fractional exponent — also not a polynomial. The rusher sees "$x$ raised to a power" and instinctively calls it a polynomial. Same kind of definitional slip behind some early CAD software bugs — calling a rational function a polynomial broke spline-fitting algorithms downstream.
Polynomial Classification Master Table — Degree × Terms
Every polynomial carries two labels at once. The grid below crosses the two axes so any polynomial you write can be placed at a glance.
Axis 1 — By Number of Terms
Terms | Name | Example | Worked-form Note |
|---|---|---|---|
1 | Monomial | $5x^3$ | A single algebraic chunk. Constants ($7$) and single-variable terms ($-2x$) both qualify. |
2 | Binomial | $x^2 - 9$ | Two terms separated by $+$ or $-$. Sum/difference of squares and cubes live here. |
3 | Trinomial | $x^2 + 5x + 6$ | Three terms after simplification. The quadratic trinomial is the textbook factoring target. |
4+ | Polynomial | $x^3 - 6x^2 + 11x - 6$ | No special name — "just a polynomial". Most cubics and higher fall here. |
Axis 2 — By Degree
Degree | Name | Example | What It Tells You |
|---|---|---|---|
0 | Constant | $7$ | A number with no variable. Graph: horizontal line. |
1 | Linear | $3x + 4$ | One step to solve. Graph: straight line. |
2 | Quadratic | $x^2 - 5x + 6$ | Quadratic formula, factoring, or completing the square. Graph: parabola. |
3 | Cubic | $2x^3 + x - 1$ | Rational Root Theorem first; Cardano's formula as fallback. Graph: S-curve. |
4 | Quartic (biquadratic) | $x^4 - 3x^2 + 2$ | Ferrari's formula or factoring as quadratic-in-$x^2$. |
5 | Quintic | $x^5 + 2x^4 - x$ | No general radical solution (Galois, 1832). Numerical methods. |
6 | Sextic | $x^6 - 1$ | Same — numerical or factoring tricks (cyclotomic). |
7 | Septic | $x^7 + x$ | Numerical methods. |
$n$ | Degree-$n$ | $x^n + \cdots$ | The general case. |
— | Zero polynomial | $p(x) = 0$ | No degree (or degree $-\infty$ by convention). |
Putting Both Labels Together
Polynomial | Degree label | Term-count label | Full classification |
|---|---|---|---|
$7$ | constant | monomial | constant monomial |
$3x$ | linear | monomial | linear monomial |
$x + 2$ | linear | binomial | linear binomial |
$x^2 - 9$ | quadratic | binomial | quadratic binomial |
$x^2 + 5x + 6$ | quadratic | trinomial | quadratic trinomial |
$2x^3$ | cubic | monomial | cubic monomial |
$x^3 + 1$ | cubic | binomial | cubic binomial |
$x^3 - 6x^2 + 11x - 6$ | cubic | polynomial (4 terms) | cubic polynomial |
$x^4 + 1$ | quartic | binomial | quartic binomial |
$x^5 + 2x^4 - x$ | quintic | trinomial | quintic trinomial |
The classification rule, in one line. Combine like terms first; then read off the highest exponent (degree) and the number of remaining terms (term-count). Two labels, applied independently, drop into the table above.
The degree label tells you which solution method to reach for; the term-count label tells you which factoring shortcut applies (difference of squares for quadratic binomials, sum/difference of cubes for cubic binomials, and so on). Read both labels every time.
Why Classifying Polynomials Matters — Summary
Types of polynomials fall into two classifications: by degree (constant, linear, quadratic, cubic, etc.) and by number of terms (monomial, binomial, trinomial, etc.).
Both classifications apply simultaneously — every polynomial has both a degree label and a term-count label.
Always combine like terms before classifying — the labels apply to the simplified form.
A polynomial requires non-negative integer exponents on every variable. Negative or fractional exponents disqualify the expression.
Real-world reach: solution method selection, computer algebra system internals, scientific computing libraries.
Three Problems to Cement Polynomial Classification
Classify $5x^4 - 3x^2 + 7$ by degree and by number of terms.
Classify the polynomial $2x^3 + 3x^3 - 4x + 5$ after simplifying. State both labels.
Decide whether each is a polynomial: (a) $x^2 + \frac{1}{x}$; (b) $3x^4 - 7$; (c) $2\sqrt{x} + 5$. Classify the ones that qualify.
If you get stuck on problem 3, recall that polynomials require non-negative integer exponents on every variable. For the surrounding theory, see cubic polynomials, standard form of a polynomial, and algebraic expressions.
Want a live Bhanzu trainer to walk through more polynomial-classification problems? Book a free demo class — online globally.
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