Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called in determinates. We can perform arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable. An example of a polynomial with one variable is x2+x-12. In this example, there are three terms: x2, x and -12.

The word polynomial is derived from the Greek words ‘poly’ means ‘many‘ and ‘nominal’ means ‘terms‘, so altogether it is said as “many terms”. A polynomial can have any number of terms but not infinite. Let’s learn about the degrees, terms, types, properties, and polynomial functions in this article.

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

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

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. An example of finding the solution of a linear equation is given below:

Example: Solve 3x — 9

Solution:

First, make the equation as 0. So,

3x — 9 = 0

⇒ 3x = 9

⇒ x = 9/3

Or, x = 3.

Thus, the solution of 3x-9 is x = 3.

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. An example to find the solution of a quadratic polynomial is given below for better understanding.

Example: Solve 3x2–6x + x3–18

Solution:

First, arrange the polynomial in the descending order of degree and equate to zero.

⇒ x3 + 3x2 -6x — 18 = 0

Now, take the common terms.

x2(x+3) — 6(x+3) =0

⇒ (x2–6)(x+3)=0

So, the solutions will be x =-3 or

x2 = 6

Or, x = ±√6

Polynomial Operations

There are four main polynomial operations which are:

• Addition of Polynomials
• Subtraction of Polynomials
• Multiplication of Polynomials

Addition of Polynomials

To add polynomials, always add the like terms, i.e. the terms having the same variable and power. The addition of polynomials always results in a polynomial of the same degree. For example,

Example: Find the sum of two polynomials: 5x3+3x2y+4xy−6y2, 3x2+7x2y−2xy+4xy2−5

Solution:

First, combine the like terms while leaving the unlike terms as they are. Hence,

(5x3+3x2y+4xy−6y2)+(3x2+7x2y−2xy+4xy2−5)

= 5x3+3x2+(3+7)x2y+(4−2)xy+4xy2−6y2−5

= 5x3+3x2+10x2y+2xy+4xy2−6y2−5

Subtraction of Polynomials

Subtracting polynomials is similar to addition, the only difference being the type of operation. So, subtract the like terms to obtain the solution. It should be noted that subtraction of polynomials also results in a polynomial of the same degree.

Example: Find the difference of two polynomials: 5x3+3x2y+4xy−6y2, 3x2+7x2y−2xy+4xy2−5

Solution:

First, combine the like terms while leaving the unlike terms as they are. Hence,

(5x3+3x2y+4xy−6y2)-(3x2+7x2y−2xy+4xy2−5)

= 5x3–3x2+(3–7)x2y+(4+2)xy-4xy2−6y2+5

= 5x3–3x2–4x2y+6xy-4xy2−6y2+5

Multiplication of Polynomials

Two or more polynomial when multiplied always result in a polynomial of higher degree (unless one of them is a constant polynomial). An example of multiplying polynomials is given below:

Example: Solve (6x−3y)×(2x+5y)

Solution:

⇒ 6x ×(2x+5y)–3y × (2x+5y) — — — — Using distributive law of multiplication

⇒ (12x2+30xy) — (6yx+15y2) — — — — Using distributive law of multiplication

⇒12x2+30xy–6xy–15y2 — — — — — — as xy = yx

Thus, (6x−3y)×(2x+5y)=12x2+24xy−15y2