Polynomials
Summarize
Definition
Let $\mathbb{R}$ be the field of real numbers. A polynomial in one variable $x$ with coefficients in $\mathbb{R}$ has the following form:
$$a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{1}x+a_{0}$$
$n$ is a non-negative integer, and $a_0,a_1,\ldots,a_n\in\mathbb{R}$ are the coefficients, with $a_n\neq 0$. Each term $a_kx^k$ is a monomial of degree $k$. We usually write a polynomial as $P(x)$ or $p(x)$. The set of all polynomials in $x$ with real coefficients is $\mathbb{R}[x]$.
The symbol $x$ is an indeterminate rather than a number to be chosen. Formally, the polynomial above is the coefficient sequence $(a_0,a_1,\ldots,a_n,0,0,\ldots)$, whose entries vanish from some index onward. Two polynomials are equal when their coefficient sequences are equal. The same definition applies over any field $\mathbb{F}$, and the resulting polynomial ring is $\mathbb{F}[x]$. The sections concerning graphs and real-variable behavior specialize to $\mathbb{R}[x]$.
The set $\mathbb{R}[x]$ is a ring under two standard operations. For two polynomials:
$$ \begin{align} P(x) &= \sum_{k=0}^{n} a_k x^k \\[6pt] Q(x) &= \sum_{k=0}^{m} b_k x^k \end{align} $$
their sum is defined by adding the coefficients of corresponding degrees:
$$(P + Q)(x) = \sum_{k=0}^{\max(n,m)} (a_k + b_k) x^k$$
The product of two polynomials is defined by the Cauchy convolution of their coefficient sequences:
$$(P \cdot Q)(x) = \sum_{k=0}^{n+m} \left( \sum_{j=0}^{k} a_j b_{k-j} \right) x^k$$
Coefficients with indices exceeding the degree of the respective polynomial are zero. Under these two operations, $\mathbb{R}[x]$ is a commutative ring with identity and an integral domain, since the product of two nonzero polynomials is never the zero polynomial.
Because $\mathbb{R}[x]$ has no zero divisors, it has the cancellation law. If $P(x)R(x)=Q(x)R(x)$ and $R(x)\neq 0$, then $(P(x)-Q(x))R(x)=0$, so $P(x)=Q(x)$.
The ring $\mathbb{R}[x]$ is closed under addition, subtraction, and multiplication.
Degree of a polynomial
The degree of a polynomial $P(x)$ is defined as the largest integer $k$ such that the coefficient $a_k$ is nonzero. This degree is denoted as $\deg P$ or $\deg P(x)$. For example, consider the polynomial:
$$P(x) = 2x^3 - 5x^2 + 3x - 7$$
$P(x)$ has degree 3, since the largest exponent appearing with a nonzero coefficient is 3. A polynomial may still be of degree 3 even if some intermediate terms are absent: the polynomial $P(x) = 4x^3 + x - 2$ is also of degree 3, despite the absence of the quadratic term.
The degree is well defined due to the requirement that $a_n \neq 0$ in the definition. The leading coefficient uniquely determines the highest-degree term, known as the leading term.
The zero polynomial, where all coefficients are zero, is the only polynomial that is not assigned a degree in the usual sense. By convention, $\deg 0 = -\infty$, a choice motivated by the requirement that the following identity remain valid even when one of the two factors is the zero polynomial:
$$\deg(P \cdot Q) = \deg P + \deg Q$$
Interpolation and degree of a polynomial
Polynomial degree controls interpolation. For distinct points $\alpha_0,\alpha_1,\dots,\alpha_n\in\mathbb{R}$ and prescribed values $\beta_0,\beta_1,\dots,\beta_n\in\mathbb{R}$, a unique polynomial $p(x)\in\mathbb{R}[x]$ of degree at most $n$ satisfies:
$$p(\alpha_i) = \beta_i \quad \forall \, i = 0, 1, \dots, n$$
Thus $n+1$ distinct interpolation nodes determine a polynomial of degree at most $n$. Its construction is polynomial interpolation, and the Lagrange interpolation formula is a standard explicit method.
Degree of a polynomial and its geometric interpretation
The degree constrains the shape of a polynomial graph. The graph of a first-degree, or linear, polynomial is a straight line of the form:
$$y = mx + q$$
In this equation $m$ is the slope and $q$ is the y-intercept. For example, the equation $y=2x+1$ is the line shown in the graph.
For a differentiable function, the derivative at a point is the slope of the tangent line at that point.
The graph of a second-degree, or quadratic, polynomial is a parabola of the form:
$$y = ax^2 + bx + c $$
The sign of $a$ determines whether the parabola opens upward or downward, the coefficients $a$, $b$, and $c$ determine the vertex, and $c$ is the y-intercept. The graph shows the parabola $y=x^2+4x-4$. It opens upward because the coefficient of $x^2$ is positive.
The graph of a third-degree, or cubic, polynomial is a cubic curve of the form:
$$y = ax^3 + bx^2 + cx + d $$
The leading coefficient $a$ determines the end orientation, the coefficients $b$ and $c$ affect the critical and inflection points, and $d$ is the y-intercept.
End behavior of polynomial
The end behavior of a polynomial is determined exclusively by its leading term, that is, the term of highest degree $a_n x^n$. As $|x|$ approaches infinity, all lower-degree terms become asymptotically negligible compared to the growth imposed by the power $x^n$.
The two end limits depend on the parity of $n$ and the sign of the leading coefficient $a_n$.
When the degree is even, the function $x^n$ is non-negative for all real values of $x$, and the end behavior is therefore symmetric: the polynomial diverges to $+\infty$ if $a_n > 0$ and to $-\infty$ if $a_n < 0$.
When the degree is odd, the power $x^n$ changes sign with $x$, yielding a non-symmetric configuration in which the two ends of the graph point in opposite directions.
| Degree $n$ | $a_n$ | $x \to -\infty$ | $x \to +\infty$ | Direction | End orientation |
|---|---|---|---|---|---|
| even | $>0$ | $+\infty$ | $+\infty$ | same | $\nwarrow$ $\nearrow$ |
| even | $<0$ | $-\infty$ | $-\infty$ | same | $\swarrow$ $\searrow$ |
| odd | $>0$ | $-\infty$ | $+\infty$ | opposite | $\swarrow$ $\nearrow$ |
| odd | $<0$ | $+\infty$ | $-\infty$ | opposite | $\nwarrow$ $\searrow$ |
The leading term determines both end limits because the ratio of every lower-degree term to $a_nx^n$ tends to zero as $|x|$ increases.
To clarify the concept further, let us consider the case in the third row with the following polynomial: $$ x^3 + 5x^2 + 5x + 1 $$
This polynomial has odd degree and a positive leading coefficient. Therefore it tends to $-\infty$ as $x\to-\infty$ and to $+\infty$ as $x\to+\infty$.
| Degree $n$ | $a_n$ | $x \to -\infty$ | $x \to +\infty$ | Direction | End orientation |
|---|---|---|---|---|---|
| odd | $>0$ | $-\infty$ | $+\infty$ | opposite | $\swarrow$ $\nearrow$ |
Every odd-degree polynomial with a positive leading coefficient has this down-to-up end behavior. Its degree and leading coefficient determine the two end limits because $a_nx^n$ dominates all lower-degree terms as $|x|$ increases.
Monomials, binomials, trinomials
A monomial is a polynomial expression comprising only one term, a constant, a single variable, or a combination of constants and variables raised to non-negative integer powers. For instance, $3x^2$ and $-5y$ are both monomials.
A binomial is a polynomial expression consisting of two terms: constants, variables, or the product of constants and variables raised to non-negative integer powers. For example, $3x + 7$ and $-2y^2 + 5y$ are both binomials.
A trinomial is a polynomial expression consisting of three terms, which can also be constants, variables, or the product of constants and variables raised to non-negative integer powers. For instance, $x^2-2x + 4$ and $3y^3 + 2y^2- y$ are both trinomials.
Sum or difference of two polynomials
The sum or difference of two polynomials of degree $n$ has degree at most $n$. It has degree less than $n$ when the leading terms cancel.
The sum or difference of the two polynomials is obtained by adding or subtracting the corresponding coefficients of the like terms.
$$ \begin{align*} P(x) + Q(x) &= (ax^n + bx^{n-1} + \ldots + z) + (px^n + qx^{n-1} + \ldots + w) \\[0.6em] &= (a+p)x^n + (b+q)x^{n-1} + \ldots + (z+w) \\[0.6em] P(x)-Q(x) &= (ax^n + bx^{n-1} + \ldots + z) - (px^n + qx^{n-1} + \ldots + w) \\[0.6em] &= (a-p)x^n + (b-q)x^{n-1} + \ldots + (z-w) \end{align*} $$
Example 1
Given two polynomials $P(x)$ and $Q(x)$, the sum $P(x) + Q(x)$ is computed as follows:
$$P(x) = x^2 + 3x - 1$$
$$Q(x) = 2x^2 - x + 5$$
The sum is given by:
$$P(x) + Q(x) = \left( x^2 + 3x - 1 \right) + \left( 2x^2 - x + 5 \right)$$
Removing the parentheses and collecting terms of equal degree we obtain:
$$ \begin{align} P(x) + Q(x) &= x^2 + 3x - 1 + 2x^2 - x + 5 \\[0.5em] &= (x^2 + 2x^2) + (3x - x) + (-1 + 5) \\[0.5em] &= 3x^2 + 2x + 4 \end{align} $$
The result of the two polynomials $P(x) + Q(x)$ is expressed as:
$$3x^2 + 2x + 4 $$
Example 2
Consider two polynomials $P(x)$ and $Q(x)$ of degree $n$. As established above, their sum or difference is a polynomial of degree at most $n$. The following example illustrates the case in which the degree strictly decreases.
$$ P(x) = 2x^2+3x-1 $$ $$ Q(x) = 2x^2-x+5 $$
The difference $P(x)-Q(x)$ is:
$$ P(x)-Q(x) = \left( 2x^2+3x-1 \right)-\left( 2x^2-x+5 \right) $$
Expanding and collecting terms of equal degree:
$$ \begin{align*} P(x)-Q(x) &= 2x^2+3x-1-2x^2+x-5 \\[0.5em] &= (2x^2-2x^2)+(3x+x)+(-1-5) \\[0.5em] &= 4x-6 \end{align*} $$
The leading terms of degree $n=2$ cancel exactly, reducing the result to a polynomial of degree $n-1=1$. This confirms that the degree of a sum or difference can be strictly less than the degree of the summands.
How to multiply two polynomials
The product of two polynomials is obtained by repeated application of the distributive property: every term of one polynomial is multiplied by every term of the other, and the resulting monomials are collected by degree. Given $P(x) = \sum_i a_i x^i$ and $Q(x) = \sum_j b_j x^j$, the product is:
$$ (P \cdot Q)(x) = \sum_{k=0}^{n+m} \left( \sum_{i=0}^{k} a_i b_{k-i} \right) x^k $$
The degree of the product satisfies $\deg(P \cdot Q) = \deg P + \deg Q$ whenever neither factor is the zero polynomial. Together with addition, multiplication endows the set $\mathbb{R}[x]$ with the structure of a commutative ring with unity and, since $\mathbb{R}$ is an integral domain, with no zero divisors. The dedicated page collects worked examples, the multivariate extension, and the connection with notable products and the FOIL method for the special case of two binomials.
How to divide two polynomials
Polynomial division asks for a quotient and a remainder. For polynomials $P(x)$ and $D(x)$ with $D(x)\neq 0$, unique polynomials $Q(x)$ and $R(x)$ satisfy:
$$P(x) = Q(x) D(x) + R(x) $$
- $Q(x)$ is the quotient of the division.
- $R(x)$ is the remainder.
- The degree of $R(x)$ is strictly less than the degree of $D(x)$
This is the polynomial division algorithm, or polynomial long division. Its remainder is zero or has degree strictly less than the degree of the divisor.
When the division between two polynomials is expressed as a reduced quotient (without explicitly showing the remainder), we obtain a rational function defined as:
$$ R(x) = \frac{P(x)}{Q(x)} $$
where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \ne 0$.
Rational equations and rational inequalities contain quotients of polynomials.
Factoring polynomials
A number $\alpha$ is said to be a root of the polynomial $P(x)$ if $P(\alpha) = 0$. The root $\alpha$ is called integer, rational, real, or complex depending on whether $\alpha$ is an integer, a rational number, a real number, or a complex number.
The existence of roots over $\mathbb{C}$ is guaranteed by the Fundamental Theorem of Algebra, which states that every non-constant polynomial with complex coefficients has at least one complex root. As a consequence, any polynomial of degree $n$ over $\mathbb{C}$ factors into exactly $n$ linear factors, counted with multiplicity. Over $\mathbb{R}$, the situation is more nuanced: real roots may not always exist, and irreducible quadratic factors with no real roots may appear in the factorization.
The general existence and uniqueness statement is developed in unique factorization of polynomials. It distinguishes irreducible factors from units and explains why a factorization is unique only up to the order of the factors and multiplication by nonzero constants.
When all roots are known, every polynomial $P(x)$ with $P(0)\ne 0$ has the following factored form:
$$ P(x) = P(0) \prod_{\rho} \left(1 - \frac{x}{\rho} \right) $$
The product has one factor for each root $\rho$, counted with multiplicity. Expanding the product and comparing coefficients gives Vieta's formulas, in which every coefficient is an elementary symmetric polynomial in the roots.
Polynomial equations
A polynomial equation is an equation of the form:
$$a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0} = 0$$
Polynomial equations are classified according to the degree of the leading term. Depending on their degree, they are referred to as linear (degree 1), quadratic (degree 2), cubic (degree 3), or of higher degree when $n > 3$.
Polynomial functions
A polynomial function is a function of the form:
$$y = a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0} $$
Each formal polynomial $P\in\mathbb{R}[x]$ has an associated function $x\mapsto P(x)$. Evaluation preserves the algebraic operations, since $(P+Q)(x)=P(x)+Q(x)$ and $(PQ)(x)=P(x)Q(x)$ for every real $x$.
Suppose that two polynomials $p(x)$ and $q(x)$ define the same function at every real value, so that:
$$ p(x) = q(x) \quad \forall \ x $$
Since $\mathbb{R}$ is infinite, the two polynomials are equal and have the same coefficients. This is the identity principle for polynomials.
The distinction between a formal polynomial and its associated function matters over finite fields. In the field $\mathbb{F}_2=\{0,1\}$, the nonzero polynomial $h(x)=x^2+x$ satisfies $h(0)=h(1)=0$. It therefore induces the same function as the zero polynomial even though their coefficient sequences differ.
Polynomial functions have the following analytical properties.
- Their domain is the entire real line $\mathbb{R}$, and they are continuous and smooth at every point, with no discontinuities, singularities, cusps, or corners.
- As a consequence of their global regularity, polynomial functions do not admit asymptotes of any kind.
- Regarding symmetry, an odd polynomial function has an inflection point at the origin $(0,0)$, while an even polynomial function attains a local maximum or minimum at $x = 0$.
The polynomial ring as a Euclidean domain
The preceding sections describe polynomials through their coefficients, graphs, and arithmetic operations. The polynomial ring $\mathbb{F}[x],$ where $\mathbb{F}$ is a field, has a divisibility theory parallel to that of the integers $\mathbb{Z}.$ For nonzero polynomials, the degree is the analogue of the absolute value of a nonzero integer.
A unit of $\mathbb{F}[x]$ is a polynomial that has a multiplicative inverse in $\mathbb{F}[x].$ If $PQ=1,$ then $\deg P+\deg Q=0,$ so both factors have degree zero. The units are therefore exactly the nonzero constant polynomials. Two polynomials that differ by a unit factor are associates. A monic polynomial has leading coefficient $1,$ and every nonzero polynomial has a unique monic associate, obtained by dividing the polynomial by its leading coefficient. In each class of nonzero associates, the monic polynomial is the chosen representative, just as the positive integer is the chosen representative in each pair $\{n,-n\}.$
Division with remainder is the basis of divisibility in $\mathbb{F}[x].$ For every pair $P,D\in\mathbb{F}[x]$ with $D\neq 0,$ unique polynomials $Q$ and $R$ satisfy $P=QD+R,$ where $R=0$ or $\deg R<\deg D.$ An integral domain with this division property is a Euclidean domain, and degree is a Euclidean function for $\mathbb{F}[x].$
At each step of the Euclidean algorithm, the new nonzero remainder has smaller degree than the preceding remainder. The algorithm therefore ends after finitely many steps.
For polynomials $F$ and $G$ that are not both zero, the Euclidean algorithm is repeated division with remainder. Its last nonzero remainder is a greatest common divisor of $F$ and $G.$ Dividing this remainder by its leading coefficient gives the unique monic greatest common divisor. Back-substitution expresses the last remainder as a polynomial combination of $F$ and $G.$ Dividing this identity by the same leading coefficient gives polynomials $S,T\in\mathbb{F}[x]$ for which Bézout's identity holds:
$$ SF+TG=\gcd(F,G) $$
Two polynomials are relatively prime when their monic greatest common divisor is $1.$ In this case Bézout's identity gives polynomials $S$ and $T$ such that $SF+TG=1.$ More generally, the combinations $AF+BG,$ where $A,B\in\mathbb{F}[x],$ are exactly the multiples of $\gcd(F,G).$
The irreducible polynomials are the analogues of the prime numbers. A polynomial of positive degree is irreducible over $\mathbb{F}$ if every factorization in $\mathbb{F}[x]$ has a unit as one of its factors. Bézout's identity implies Euclid's lemma, according to which an irreducible polynomial that divides a product divides one of its factors. Induction on the degree proves the existence of a factorization into irreducible polynomials, while Euclid's lemma proves its uniqueness up to the order of the factors and multiplication by units. If each irreducible factor is monic, the unit factor is the leading coefficient $a,$ and the factorization has the form:
$$ P(x)=aP_1(x)^{e_1}\cdots P_r(x)^{e_r} $$
Here $P_1,\ldots,P_r$ are distinct monic irreducible polynomials, and $e_1,\ldots,e_r$ are positive integers. Every Euclidean domain is a principal ideal domain, and every principal ideal domain is a unique factorization domain. Thus $\mathbb{F}[x]$ has both properties.
The corresponding notions in $\mathbb{Z}$ and $\mathbb{F}[x]$ are summarized in the following table.
| Integers $\mathbb{Z}$ | Polynomials $\mathbb{F}[x]$ |
|---|---|
| Euclidean function $\lvert n\rvert$ | Euclidean function $\deg P$ |
| units $\pm 1$ | nonzero constant polynomials |
| positive associate representatives | monic associate representatives |
| prime numbers | irreducible polynomials |
| remainder with $\lvert r\rvert < \lvert d\rvert$ | remainder with $\deg r < \deg d$ |
| $\gcd$ and Bézout's identity | $\gcd$ and Bézout's identity |
| unique factorization into primes | unique factorization into irreducibles |
The field $\mathbb{F}$ determines which polynomials are irreducible. Over $\mathbb{C},$ every irreducible polynomial is linear by the fundamental theorem of algebra. Over $\mathbb{R},$ the irreducible polynomials are precisely the linear polynomials and the quadratic polynomials with negative discriminant. Over $\mathbb{Q},$ Eisenstein's criterion with the prime $2$ shows that $x^n-2$ is irreducible for every positive integer $n,$ so the degrees of the irreducible polynomials have no upper bound. Each of the rings $\mathbb{C}[x],$ $\mathbb{R}[x],$ and $\mathbb{Q}[x]$ has infinitely many monic irreducible polynomials. These distinctions, together with the relation between $\mathbb{Z}[x]$ and $\mathbb{Q}[x],$ are discussed in unique factorization of polynomials.