Polynomials

What is a Polynomial?

Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division (No division operation by a variable). Based on the number of terms present in the expression, it is classified as monomial, binomial, and trinomial. Examples of constants, variables and exponents are as follows:

  • Constants. Example: 1, 2, 3, etc.
  • Variables. Example: g, h, x, y, etc.
  • Exponents: Example: 5 in x5 etc.

Standard Form of a Polynomial

P(x) = anxn + an-1xn-1 +an-2xn-2 + ………………. + a1x + a0   

Where an, an-1, an-2, ……………………, a1, a0 are called coefficients of xn, xn-1, xn-2, ….., x and constant term respectively and it should belong to real number (⋲ R).

Notation

The polynomial function is denoted by P(x) where x represents the variable. For example,

P(x) = x2-5x+11

If the variable is denoted by a, then the function will be P(a)

Degree of a Polynomial

The degree of a polynomial is defined as the highest exponent of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the largest exponent is called a degree of the polynomial.

Polynomial Degree Example
Zero Polynomial Not Defined 6
Constant 0 P(x)  = 6
Linear Polynomial 1 P(x) = 3x+1
Quadratic Polynomial 2 P(x) = 4x2+1x+1
Cubic Polynomial 3 P(x) = 6x3+4x2+3x+1
Quartic Polynomial 4 P(x) = 6x4+3x3+3x2+2x+1

Example: Find the degree of the polynomial P(x) = 6s4+ 3x2+ 5x +19

Solution:

The degree of the polynomial is 4 as the highest power of the variable 4.

Terms of a Polynomial

The terms of polynomials are the parts of the expression that are generally separated by “+” or “-” signs. So, each part of a polynomial in an expression is a term. For example, in a polynomial, say, 2x2 + 5 +4, the number of terms will be 3. The classification of a polynomial is done based on the number of terms in it.

Polynomial Terms Degree
P(x) = x3-2x2+3x+4 x3, -2x2, 3x and 4 3

Types of Polynomials

Depending upon the number of terms, polynomials are divided into the following categories:

  • Monomial
  • Binomial
  • Trinomial
  • Polynomial containing 4 terms (Quadronomial) 
  • Polynomial containing 5 terms (pentanomial ) and so on …

These polynomials can be combined using addition, subtraction, multiplication, and division but is never divided by a variable. A few examples of Non Polynomials are: 1/x+2, x-3

Monomial

A monomial is an expression which contains only one term. For an expression to be a monomial, the single term should be a non-zero term. A few examples of monomials are:

  • 5x
  • 3
  • 6a4
  • -3xy

Binomial

A binomial is a polynomial expression which contains exactly two terms. A binomial can be considered as a sum or difference between two or more monomials. A few examples of binomials are:

  • – 5x+3,
  • 6a4 + 17x
  • xy2+xy

Trinomial

A trinomial is an expression which is composed of exactly three terms. A few examples of trinomial expressions are:

  • – 8a4+2x+7
  • 4x2 + 9x + 7
Monomial Binomial Trinomial
One Term Two terms Three terms
Example: x, 3y, 29, x/2 Example: x2+x, x3-2x, y+2 Example: x2+2x+20

Properties

Some of the important properties of polynomials along with some important polynomial theorems are as follows:

Property 1: Division Algorithm

If a polynomial P(x) is divided by a polynomial G(x) results in quotient Q(x) with remainder R(x), then,

P(x) = G(x) Q(x) + R(x)

Where R(x)=0 or the degree of R(x) < the degree of G(x)

Property 2: Bezout’s Theorem

Polynomial P(x) is divisible by binomial (x – a) if and only if P(a) = 0.

Property 3: Remainder Theorem

If P(x) is divided by (x – a) with remainder r, then P(a) = r.

Property 4: Factor Theorem

A polynomial P(x) divided by Q(x) results in R(x) with zero remainders if and only if Q(x) is a factor of P(x).

Property 5: Intermediate Value Theorem

If P(x) is a polynomial, and P(x) ≠ P(y) for (x < y), then P(x) takes every value from P(x) to P(y) in the closed interval [x, y].

Property 6

The addition, subtraction and multiplication of polynomials P and Q result in a polynomial where,

Degree(P ± Q) ≤ Degree(P or Q)

Degree(P × Q) = Degree(P) + Degree(Q)

Property 7

If a polynomial P is divisible by a polynomial Q, then every zero of Q is also a zero of P.

Property 8

If a polynomial P is divisible by two co-prime polynomials Q and R, then it is divisible by (Q • R).

Property 9

If P(x) = a0 + a1x + a2x2 + …… + anxn is a polynomial such that deg(P) = n ≥ 0 then, P has at most “n” distinct roots.

Property 10: Descartes’ Rule of Sign

The number of positive real zeroes in a polynomial function P(x) is the same or less than by an even number as the number of changes in the sign of the coefficients. So, if there are “K” sign changes, the number of roots will be “k” or “(k – a)”, where “a” is some even number.

Property 11: Fundamental Theorem of Algebra

Every non-constant single-variable polynomial with complex coefficients has at least one complex zero.

Property 12

If P(x) is a polynomial with real coefficients and has one complex zero (x = a – bi), then x = a + bi will also be a zero of P(x). Also, x2 – 2ax + a2 + b2 will be a factor of P(x).

Polynomial Equations

Polynomial equations are those expressions which are made up of multiple constants and variables. The standard form of writing a polynomial equation is to put the highest degree first and then, at last, the constant term. An example of a polynomial equation is:

0 = a4 +3a3 -2a2 +a +1

Polynomial Functions

A polynomial function is an expression constructed with one or more terms of variables with constant exponents. If there are real numbers denoted by a, then function with one variable and of degree n can be written as:

f(x) = a0xn + a1xn-1 + a2xn-2 + ….. + an-2x2 + an-1x + an

Solving Polynomials

Any polynomial can be easily solved using basic algebra and factorization concepts. While solving the polynomial equation, the first step is to set the right-hand side as 0. The explanation of a polynomial solution is explained in two different ways:

  • Solving Linear Polynomials
  • Solving Quadratic Polynomials

Solving Linear Polynomials

Getting the solution of linear polynomials is easy and simple. First, isolate the variable term and make the equation as equal to zero. Then solve as basic algebra operation.

Solving Quadratic Polynomials

To solve a quadratic polynomial, first, rewrite the expression in the descending order of degree. Then, equate the equation and perform polynomial factorization to get the solution of the equation.