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What is a Polynomial?
In the field of Mathematics, a polynomial is an equation/ expression that consists of variables and coefficients, that involve only the mathematical operations of addition, subtraction, multiplication, division and integer exponents which are non-negative numbers.
Polynomial is a concept explored in the Algebra area of mathematics. Now onto the basic terms that build a polynomial before we go through the definition of a polynomial and its many kinds.
Example of a Polynomial
3x
2y
10xy
15x + 10y, this is an example of a Linear Polynomial.
x2 – 3x + 16, this is an example of a Quadratic Polynomial.
16×3 + 14×3 + 2x + 2, this is an example of a Cubic Polynomial.
10×4 + 4×4 + 2×3 + 2, this is an example of a Quartic Polynomial.
Types of a Polynomial
In Mathematics, the use of letters or an alphabet to denote variables has been very common. In a general formula for a polynomial of one variable, the ‘a’ denotes the constants and the ‘x’ denoted a variable.
Monomial
In the field of Mathematics, a Monomial is a type of polynomial which has only one term.
For Example- 2x, 3x, 4x, 5y, 4xy, 10bc, etc.
Binomial
In the field of Mathematics, a Binomial is a type of polynomial that consists of only two terms or two monomials.
For Example- 2x + 3x, 4x + 5y, 4x + y, 10b + c, etc.
Trinomial
In the field of Mathematics, a Trinomial is a type of polynomial that consists of only three terms or three monomials.
For Example- 5x + 8y + 14z, 5y + 4x + z, 10b + c + 20d, etc.
Uses of a Polynomial
Polynomials play a very important role when it is about equation solving in the field of Mathematics and Science.
- They are used to form Polynomial Equations that encode a wide range of problems, starting from basic word problems to scientific equation solving problems in Chemistry, Physics and Biology.
- They are used to define Polynomial Functions that are seen in Basic Chemistry, Physics, Social Sciences, Economics, etc.
- Polynomials are also used in Calculus and Analysis of Numeric Data to find the approximation of various other functions.
Degree of a Polynomial
In the field of Mathematics, the Degree of Polynomial is the highest degree of its terms, when the polynomial is expressed in a linear form of monomials (called a canonical form). The Degree of a Polynomial is the sum of the exponents of the variables that are present in the given equation.
Degree of 0 – Called as a Constant Polynomial.
Degree of 1 – Called as a Linear Polynomial.
Degree of 2 – Called as a Binomial Polynomial.
Degree of 3 – Called as a Trinomial Polynomial.
Degree of 4 – Called as a Quartic or Biquadratic Polynomial.
How to find the Degree of a Polynomial?
Consider, x2 – 2×4 + 3×3 + 5×6 is a given Polynomial.
We must observe all the terms of the given polynomial, the alphabet or variable with the maximum number of power (the highest number) is the degree of that given polynomial.
a – b2 = Degree of Polynomial is 2 as the maximum power is 2.
x4 – y3 = Degree of Polynomial is 4 as the maximum power is 4.
a4 + b3 = Degree of Polynomial is 4 as the maximum power is 4.
x4 – y7 = Degree of Polynomial is 7 as the maximum power is 7.
p10 – q2 = Degree of Polynomial is 10 as the maximum power is 10.
x4y4 + x3 = Degree of Polynomial is 8 as the maximum power is 8.
x14 – y7 = Degree of Polynomial is 14 as the maximum power is 14.
p11 – q2 = Degree of Polynomial is 11 as the maximum power is 11.
a4b2 + c3 = Degree of Polynomial is 6 as the maximum power is 6.
Some Important Algebraic Identities Related to Polynomials
ax + b = 0 (Linear).
ax2 + bx + c = 0 (Quadratic).
ax3 + bx2 + cx + d = 0 (Cubic).
(a + b)2 = a2 + b2 + 2ab.
(a – b)2 = a2 + b2 – 2ab.
(a2 – b2) = (a + b) (a – b).
(x – a) (x – b) = x2 + (a + b) x – ab.
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac.
(a + b)3 = a3 + b3 + 3a2b + 3ab2.
(a – b)3 = a3 – b3 – 3a2b + 3ab2.
a3 + b3 = (a + b) (a2 – ab + b2).
Conclusion
A polynomial is a mathematical expression that contains variables, coefficients, and the addition, subtraction, multiplication, and non-negative integer exponents operations.
To solve difficult time-consuming equations that have a higher degree, knowing polynomials is very important.
One must know all the formulae of this very topic to make it easy when it comes to equation solving in school, college and university examinations and not only school and college examinations but also competitive examinations.
Polynomials are the basics of calculus and a lot of other concepts in various other subjects, so it is a must to know this topic.
a3 – b3 = (a – b) (a2 + ab + b2).
