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Integer Polynomial — Definition, Formula & Examples

An integer polynomial is a polynomial whose coefficients are all integers. For example, 3x2−5x+73x^2 - 5x + 7 is an integer polynomial, but 12x2+x\frac{1}{2}x^2 + x is not.

A polynomial p(x)=anxn+an−1xn−1+⋯+a1x+a0p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 is called an integer polynomial (or a polynomial over Z\mathbb{Z}) if every coefficient ai∈Za_i \in \mathbb{Z} for i=0,1,…,ni = 0, 1, \ldots, n. The set of all such polynomials is denoted Z[x]\mathbb{Z}[x].

Key Formula

p(x)=anxn+an−1xn−1+⋯+a1x+a0,ai∈Zp(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, \quad a_i \in \mathbb{Z}
Where:
  • aia_i = Each coefficient, required to be an integer
  • nn = The degree of the polynomial (a non-negative integer)
  • xx = The indeterminate (variable)

How It Works

To determine whether a polynomial is an integer polynomial, check each coefficient: if every one is an integer (positive, negative, or zero), the polynomial belongs to Z[x]\mathbb{Z}[x]. Integer polynomials are closed under addition, subtraction, and multiplication, meaning performing these operations on two integer polynomials always produces another integer polynomial. However, dividing one integer polynomial by another may yield non-integer coefficients, so Z[x]\mathbb{Z}[x] is a ring but not a field. Results like the Rational Root Theorem and Eisenstein's criterion apply specifically to integer polynomials, making them central to factoring and irreducibility questions in algebra and number theory.

Worked Example

Problem: Determine whether p(x)=4x3−x2+6p(x) = 4x^3 - x^2 + 6 is an integer polynomial, and find all possible rational roots using the Rational Root Theorem.
Check coefficients: The coefficients are 4, −1, 0, and 6. All are integers, so p(x)∈Z[x]p(x) \in \mathbb{Z}[x].
a3=4,  a2=−1,  a1=0,  a0=6a_3 = 4,\; a_2 = -1,\; a_1 = 0,\; a_0 = 6
Apply the Rational Root Theorem: For an integer polynomial, any rational root pq\frac{p}{q} (in lowest terms) has pp dividing the constant term and qq dividing the leading coefficient.
Divisors of 6:±1,±2,±3,±6Divisors of 4:±1,±2,±4\text{Divisors of } 6: \pm1, \pm2, \pm3, \pm6 \qquad \text{Divisors of } 4: \pm1, \pm2, \pm4
List candidates: Form all ratios pq\frac{p}{q} and reduce to get the possible rational roots.
±1,  ±2,  ±3,  ±6,  ±12,  ±32,  ±14,  ±34\pm1,\; \pm2,\; \pm3,\; \pm6,\; \pm\tfrac{1}{2},\; \pm\tfrac{3}{2},\; \pm\tfrac{1}{4},\; \pm\tfrac{3}{4}
Answer: Yes, p(x)p(x) is an integer polynomial. Its possible rational roots are ±1,±2,±3,±6,±12,±32,±14,±34\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}.

Why It Matters

Integer polynomials are the starting point for the Rational Root Theorem, Eisenstein's irreducibility criterion, and much of algebraic number theory. In abstract algebra courses, Z[x]\mathbb{Z}[x] serves as a fundamental example of a unique factorization domain (UFD), connecting polynomial algebra to ring theory.

Common Mistakes

Mistake: Assuming that dividing two integer polynomials always yields another integer polynomial.
Correction: Z[x]\mathbb{Z}[x] is closed under addition, subtraction, and multiplication, but not division. For instance, x2+12=12x2+12∉Z[x]\frac{x^2 + 1}{2} = \frac{1}{2}x^2 + \frac{1}{2} \notin \mathbb{Z}[x].