What Is a Polynomial?

A polynomial is an algebraic expression in which terms are separated using the "+" and "-" operators. In this guide, you will learn more about the definition of a polynomial and its properties.

What Is a Polynomial?

A polynomial is a kind of expression. An expression is a mathematical statement without an equal-to sign \((=)\).

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Step by step guide to polynomial

A polynomial is a type of algebraic expression in which the exponents of all variables must be a whole number. The power of variables in any polynomial must be a non-negative integer. A polynomial contains constants and variables, but we can not perform division operations by a variable in polynomials.

The following image shows all the terms in a polynomial:

Terms of a polynomial

Polynomial terms are defined as parts of an expression that are separated by the operators \(+\) or \(-\). For example, the polynomial expression \(5x^3-\:4x^2+\:8x\:-12\) consists of four terms.

Degree of a polynomial

The highest or greatest power of a variable in a polynomial is known as the degree of the polynomial. The degree is used to determine the maximum number of solutions of a polynomial equation.

Note: The degree of a polynomial with more than one variable is equal to the sum of the power of the variables in it.

Types of polynomials

Polynomials can be categorized based on their degree and power. Based on the number of terms, there are mainly three types of polynomials listed below:

  • Monomials
  • Binomials
  • Trinomials

A monomial is a type of polynomial with a single term. For example, \(x\), \(-4xy\), and \(7z^2\). A binomial is a type of polynomial that has two terms. For example, \(x + 3\), \(y^2+5\), and \(2x^3- 6\). While a trinomial is a type of polynomial that has three terms. For example \(3x^2+ 7x -5\), \(x + y + z\), and \(4x + y- 8\).

However, based on the degree of the polynomial, polynomials can be classified into \(4\) main types:

  • Zero polynomial
  • Constant polynomial
  • Linear polynomial
  • Quadratic polynomial
  • Cubic polynomial

A constant polynomial is defined as a polynomial whose degree is zero. Any constant polynomial with coefficients equal to zero is defined as a zero polynomial. For example, \(2, 4,\) or \(8\). Polynomials whose degree of the polynomial is 1 are called linear polynomials. For example, \(x + y-3\). Polynomials with \(2\) as the degree of the polynomial are called quadratic polynomials. For example, \(2n^2 – 8\). Polynomials with \(3\) as the degree of the polynomial are called cubic polynomials. For example, \(9m^3-mn+n^2-2\).

Properties of polynomials

A polynomial expression has terms connected by the addition or subtraction operators. Depending on the type of polynomial and the operation performed, there are different properties and theorems about polynomials. Some of these are given below:

Theorem 1:  If \(A\) and \(B\) are two given polynomials then,

  • \(deg⁡ (A ± B) ≤\) \(max\: (deg⁡ A, deg ⁡B\)), with the equality, if \(deg⁡ A ≠ deg\) \(⁡B\)
  • \(deg⁡ (A⋅B) = deg⁡ A + deg⁡ B\)

Theorem 2: Given polynomials \(A\) and \(B ≠ 0\), there are unique polynomials \(Q\) (quotient) and \(R\) (residue) such that, \(A = BQ + R\) and \(deg R < deg B\).

Theorem 3 (Bezout’s Theorem): Polynomial \(P(x)\) is divisible by binomial \(x − a\), if and only if \(P(a) = 0\).

Theorem 4: If the polynomials of \(P\) are divisible by the polynomials of \(Q\), every zero of \(Q\) is also a zero of \(P\).

Theorem 5: Polynomial \(P(x)\) of degree \(n > 0\) has a unique representation of the form \(P(x) = k(x – x_1)(x -x_2)…(x -x_n)\), where \(k ≠ 0\) and \(x_1,…,x_n\) are complex numbers, not necessarily distinct. Therefore, \(P(x)\) has at most \(deg ⁡P = n\) different zeros.

Theorem 6: A polynomial of \(n\)-th degree has exactly \(n\) complex roots along with their multiplicities.

Theorem 7: If a polynomial \(P\) is divisible by two coprime polynomials \(Q\) and \(R\), then it is divisible by \(Q⋅R\).

Theorem 8: A real polynomial \(P(x)\) has a unique factorization (up to the order) of the form, \(P\left(x\right)=\left(x\:-\:r_1\right)…\left(x\:-\:r_k\right)\left(x^2\:-\:p_1x\:+\:q_1\right)..\left(x^2\:-\:p_lx\:+\:q_l\right)\),

where \(r_i\) and \(p_j\), \(q_j\) are real numbers with \(p_i^2 < 4q_i\) and \(k + 2l = n\).

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