Polynomials: Definition, Types and Examples

Polynomial is an expression which comprises of variables and coefficient involving the basic arithmetic operations of addition, subtraction, and Multiplication and the exponential negative exponential of variables. 

The origin of the word polynomial is from the Greek language. ’Poly – many’ and ‘nomial’ – terms. So as a whole the word polynomial refers to many terms/multiple terms.

Also Read:- Class 10 Degree of polynomial

There are several features of a polynomial: -

• Notation
• Degree
• Terms
• Types

Notation: - Polynomials are denoted by p(x)where x is the variable and p is the denotation.

Example: - 5x2+3x-2

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Degree of an Polynomial

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Degree of a polynomial is defined as the highest power of a monomial present in a polynomial expression. So, a polynomial equation with consists of the highest exponential power is known as the degree of a polynomial.

Polynomials are categorized into 5 categories based on their degree: -

The video below explains this:

Polynomials Detailed Video Explanation:

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Terms of a Polynomial 

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A polynomial is made up of 2 terms. Mainly the constant and the variable.

Constant: - The term whose value is pre-defined and does not change in any case is known as a constant. 

Example: - 2,6,8

Variable: - The term whose value is not predefined and is bound to change with the requirements of the problem.

Example: - x,y

Also Read:- Complex Numbers and Quadratic Equations

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Types of Polynomials

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There are Constituently 3-types of polynomials: -

Monomial

A polynomial that comprises a single term/mono term is known as a monomial expression. The single term in this monomial expression should be non zero.

Example: - 7x

Binomial 

A polynomial expression that consists of precisely two terms is known as Binomial expression. It can also be understood as an arithmetic operation between two monomial expressions.

Example: - 3x+8, 5x-9

Trinomial 

A polynomial that consists of exactly 3 terms is known as a trinomial expression. It can also be interpreted as an arithmetic operation between one Binomial and Monomial term or an operation between 3 monomial terms.

Example: - 4x2+3x+1 

• Zeroes of Polynomial

Zeros of a polynomial are known as the points where the polynomial becomes zero as a whole. A polynomial having a value zero is called a zero polynomial. Highest power associated with variable x is the degree of a polynomial.

Also Read: 

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Solving Polynomial Equations

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Polynomial equations require basic knowledge of Arithmetic and Algebra operations and some basic properties that are specific to polynomial equations.

Algebraic Properties of Polynomials: -

Division Algorithm-If a polynomial P(x) is divided by a polynomial f(x) results in quotient Q(x) with the remainder an (x), then,

P(x) = f(x) • Q(x) + a(x)

Bezout’s Theorem-

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

Remainder Theorem-

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

Factor Theorem-

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

Intermediate Value Theorem-

If P(x) is a polynomial, and P(x) is not equal to P(y) for (x < y), then P(x) takes every value from P(x) to P(y) in the closed interval [x, y].

Arithmetic property-

The addition, subtraction, and multiplication of polynomials P and Q give an output in the form of a polynomial where,

Degree(P ± Q) ≤ Degree(P or Q)

Degree(P × Q) = Degree(P) + Degree(Q)

Fundamental Theorem of Algebra-

Every non-constant single-variable polynomial which has complex coefficients has at least one complex root

Also Read:- Roots of Polynomials

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Solving Quadratic polynomial equations

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Quadratic polynomial equations are solved using polynomial factorization.

The equation has to be rearranged according to the degree so it becomes easy to factorize it further.

Equating the equation to zero is a necessary step as well.

Example:   Arrange the polynomial in the decreasing order of degree and then make them equal 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 and

x² = 6

Or, x = √6

Polynomial Operations: -

There are mainly 4 polynomial operations: -

• Addition of Polynomials – Polynomials of the same degree and variable are added. The result is also a polynomial
• Subtraction of Polynomials – Polynomials of the same degree and variable are subtracted. The result is also a polynomial
• Multiplication of Polynomials- Two or more polynomial when multiplied always result in the form of a polynomial of higher degree

Example – 

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

Division of Polynomials – 

1. Write the polynomial expression in descending order.
2. Check the highest power and divide the terms by it.
3. Use result from step 2 as the division symbol.
4. Now subtract it and bring it down the next term.
5. Repeat steps 2 to 4 until you don’t have any more terms to carry down.
6. You get the final answer, including the remainder, which will be in the fraction form (last subtract term).

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Sample Linear Polynomial Problem with a solution

Question: (2x-6) Solve this polynomial for the value of x (2 marks)