A polynomial is an algebraic expression built from variables, constants, and non-negative integer exponents, combined using addition, subtraction, and multiplication. The general form is $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where the highest exponent $n$ is the polynomial's degree. Polynomials are classified two ways — by the number of terms (monomial, binomial, trinomial) and by degree (linear, quadratic, cubic, and higher).
The key formulas you'll meet most often are the algebraic identities — $(a+b)^2$, $(a-b)^2$, $(a+b)(a-b)$, the cubic identities, and the Factor and Remainder theorems.
Quick Reference
Field | Value |
|---|---|
Definition | Algebraic expression of variables, constants, and non-negative integer exponents combined by $+$, $-$, $\times$ |
General form | $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ |
Types by terms | Monomial (1), Binomial (2), Trinomial (3), Polynomial (4+) |
Types by degree | Linear (1), Quadratic (2), Cubic (3), Quartic (4), Quintic (5), … |
Key formulas | $(a \pm b)^2$, $(a+b)(a-b)$, $(a \pm b)^3$, $a^3 \pm b^3$, Factor theorem, Remainder theorem |
Used in | Algebra, calculus, computer graphics (Bezier curves), error-correcting codes, cryptography |
What Is A Polynomial?
A polynomial is the simplest kind of algebraic expression that still does interesting work. It is made of terms, and each term is a constant multiplied by a variable raised to a whole-number power. Add or subtract terms together and you have a polynomial.
The expression $5x^3 - 2x^2 + 7x - 4$ is a polynomial in the variable $x$. Each term has three parts: a coefficient (the number out front), a variable, and an exponent that must be a non-negative integer. The degree of the whole polynomial is the highest exponent that appears — here, $3$.
The expression $\sqrt{x} + 1$ is not a polynomial. The square root is the same as $x^{1/2}$, and that exponent is not a whole number. Likewise $\frac{1}{x}$ is not a polynomial — it hides $x^{-1}$, a negative exponent.
The Types of Polynomials
There are two ways to sort polynomials, and you'll need both.
By Number of Terms
A polynomial with one term is a monomial: $7x^2$, $-3$, $\frac{1}{2}xy$. Two terms make a binomial: $x^2 + 5$, $3x - 7$. Three terms make a trinomial: $x^2 + 5x + 6$. Four or more terms — the word polynomial covers everything beyond three terms, and the names stop being useful.
Name | Number of terms | Example |
|---|---|---|
Monomial | 1 | $4x^2$ |
Binomial | 2 | $x^2 - 9$ |
Trinomial | 3 | $x^2 + 5x + 6$ |
Polynomial | 4 or more | $x^4 + 2x^3 - x^2 + 7x - 1$ |
By Degree
The degree is the highest exponent in the polynomial. The names map directly to it.
Name | Degree | Example | Where you see it |
|---|---|---|---|
Constant | 0 | $7$ | Fixed quantities |
Linear | 1 | $3x + 2$ | Lines, simple rates |
Quadratic | 2 | $x^2 - 4x + 3$ | Parabolas, projectile motion |
Cubic | 3 | $x^3 + 2x^2 - x + 5$ | Volume problems, the 1535 duel |
Quartic | 4 | $x^4 - 16$ | Optics, optimisation |
Quintic | 5 | $x^5 - x + 1$ | The frontier — no general formula exists |
A constant polynomial like $7$ has degree $0$ because $7 = 7x^0$. The zero polynomial — just $0$ — is a special case; it has no defined degree.
The Key Formulas
Most working with polynomials reduces to a handful of identities. Memorise the shapes; the rest follows.
Algebraic identities (binomial expansions):
$$(a + b)^2 = a^2 + 2ab + b^2$$
$$(a - b)^2 = a^2 - 2ab + b^2$$
$$(a + b)(a - b) = a^2 - b^2$$
$$(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$$
$$(a - b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3$$
$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$
$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$
Factor theorem. If $P(c) = 0$, then $(x - c)$ is a factor of $P(x)$.
Remainder theorem. When you divide $P(x)$ by $(x - c)$, the remainder is $P(c)$.
The quadratic formula. For $ax^2 + bx + c = 0$,
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
For the cubic and quartic, formulas exist but are unwieldy — that's where the 1535 story below picks up.
Working With Polynomials - Three Worked Examples
Example 1: Subtracting two polynomials (the wrong path first)
Subtract $(2x^2 - 3x + 4)$ from $(5x^2 + x - 1)$.
The instinct here is to write the two expressions in a row and subtract term by term. Try it that way first:
$$5x^2 + x - 1 - 2x^2 - 3x + 4$$
Combine: $5x^2 - 2x^2 = 3x^2$, $x - 3x = -2x$, $-1 + 4 = 3$. Answer: $3x^2 - 2x + 3$.
Now check that against what subtraction actually means. We are taking all of $(2x^2 - 3x + 4)$ away from $(5x^2 + x - 1)$. The minus sign has to apply to every term of the second polynomial, including the $-3x$ and the $+4$. Without parentheses, only the $2x^2$ got the minus.
The correct way:
$$(5x^2 + x - 1) - (2x^2 - 3x + 4)$$
$$= 5x^2 + x - 1 - 2x^2 + 3x - 4$$
$$= 3x^2 + 4x - 5$$
Final answer: $3x^2 + 4x - 5$.
The rusher gets the first version every time. Slowing down to write the parentheses is the difference.
Example 2: Factoring a quadratic
Factor $x^2 + 5x + 6$.
Find two numbers whose product is $6$ and whose sum is $5$. Those numbers are $2$ and $3$.
$$x^2 + 5x + 6 = (x + 2)(x + 3)$$
Final answer: $(x + 2)(x + 3)$.
Example 3: Finding the degree of a complicated polynomial
What is the degree of $4x^2 y^3 - 7xy + 2$?
For a multivariable term, the degree is the sum of the exponents in that term. The first term has degree $2 + 3 = 5$, the second has $1 + 1 = 2$, the third is a constant of degree $0$.
Final answer: The polynomial's degree is $5$ — the highest among its terms.
The Mathematicians Who Shaped Polynomials
The story of polynomials is the story of how mathematicians fought, sometimes literally, to extend the formula. The Babylonians were already solving quadratic problems by 2000 BCE, and Diophantus systematised polynomial notation in 3rd-century Alexandria. The drama begins with the cubic.
In early-1500s Bologna, Scipione del Ferro (1465–1526, Italy) found a method to solve a special form of cubic — and kept it secret. He passed it on to his student Antonio Maria Fior on his deathbed. Niccolò Tartaglia (1500–1557, Italy) - a self-taught mathematician with a speech impediment from a French sword wound to the jaw — independently rediscovered the method in 1535 and beat Fior in a public mathematical duel. Each had set the other thirty cubic problems to solve in forty days; Tartaglia solved all thirty in two hours.
The Milanese physician and gambler Gerolamo Cardano (1501–1576, Italy) heard about it and convinced Tartaglia to share the method, swearing under oath never to publish it. Cardano then learned that del Ferro had got there first, decided the oath no longer bound him, and published the method in his 1545 book Ars Magna — crediting both men, but breaking the oath all the same. The two never spoke again. The same investigation forced sixteenth-century mathematicians to take the square roots of negative numbers seriously for the first time, which is how complex numbers entered mathematics.
The story doesn't end there. Évariste Galois (1811–1832, France) — who died at twenty in a duel over a love affair — proved that no general formula in radicals exists for polynomials of degree five or higher. The night before the duel, he wrote up his proof in a letter to a friend; that letter founded the field now called Galois theory.
Polynomials live well beyond schoolbooks. Bezier curves in fonts, animation, and graphic design are cubic polynomials in disguise. GPS satellites use polynomial Reed–Solomon codes to recover from corrupted signals. RSA cryptography, the system that secures your bank traffic, depends on the algebra of polynomials over finite fields.
The simple expression you started with $-$ $5x^3 - 2x^2 + 7x - 4$ $-$ is one rung of a very long ladder.
Common Polynomial Mistakes
These are the four errors that wreck more polynomial problems than every other mistake combined.
Mistake 1: Subtracting without parentheses.
Where it slips in: Whenever the problem says "subtract polynomial A from polynomial B" and the student writes the two expressions in a row.
Don't do this: $5x^2 + x - 1 - 2x^2 - 3x + 4$, treating only the leading term of the second polynomial as negative.
The correct way: Bracket the second polynomial first: $(5x^2 + x - 1) - (2x^2 - 3x + 4)$. Distribute the minus sign to every term inside before combining.
Mistake 2: Adding the exponents to the coefficients.
Where it slips in: Multiplying two monomials like $3x^2 \cdot 4x^3$.
Don't do this: Write $7x^5$ — adding $3 + 4$ to get the coefficient.
The correct way: Multiply coefficients, add exponents: $3 \times 4 = 12$ and $x^2 \cdot x^3 = x^{2+3} = x^5$, so the answer is $12x^5$. The memorizer who learned "add exponents" without learning "multiply coefficients" hits this every time.
Mistake 3: Confusing the degree with the number of terms.
Where it slips in: Reading a long polynomial like $x^4 + 3x^2 - 5$ and announcing the degree as $3$ because there are three terms.
Don't do this: Count the terms.
The correct way: Find the highest exponent among the variables. For $x^4 + 3x^2 - 5$, the highest exponent is $4$, so the degree is $4$. Number of terms describes the type (binomial, trinomial); degree is a separate property.
Mistake 4: Failing to combine like terms.
Where it slips in: After expanding a product or simplifying a sum, the answer still has $3x$ and $5x$ as separate terms.
Don't do this: Leave the expression as $3x + 5x + 2$.
The correct way: Combine: $3x + 5x = 8x$, so the simplified form is $8x + 2$. The silent understander often skips this step because it feels obvious — and then loses marks because the unsimplified answer is technically incomplete.
The real-world version of Mistake 1 is one of the most expensive transcription errors in the history of spaceflight. In 1962, NASA's Mariner 1 probe was destroyed seventy-four seconds after launch. The cause was traced to a missing overbar in a handwritten guidance equation — the bar over a velocity term, indicating a smoothed average, had been omitted in transcription.
The trajectory polynomial in the on-board computer therefore reacted to instantaneous velocity instead of the average, and the rocket veered off course. A single missing mark in a polynomial. The cost in 1962 dollars was around $80 million.
Try These next
Now try factoring $x^2 + 7x + 12$ on your own. If you get stuck, come back to Example 2 and find the two numbers whose product and sum match. From there, work through Mistakes 1–4 by writing your own polynomial in each shape and watching where the trap is.
If your child is comfortable with single-variable polynomials, the natural next step is multivariable polynomials and polynomial functions of higher degree — and after that, the algebra of polynomial roots that took mathematicians 250 years to settle. At Bhanzu, trainers connect each polynomial identity to where it shows up next — in calculus, in coordinate geometry, in coding theory — so the formulas don't sit isolated in a chapter.
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