What is a polynomial in Maths?
Polynomial is an expression that contains variables, coefficients, or constant terms with addition, subtraction, etc. signs.
Examples of polynomial are 8n³, 5z-z²+12, 6s³+4s²-5s-1, 89, etc.
Classification of Polynomials
Polynomials are classified according to the number of terms & according to their degree. Let's discuss its classifications-
Types of polynomials based on terms:-
Here we are classifying them according to the number of terms present in the polynomial.
Monomial: If the number of terms is one then the expression is known as a monomial. Examples- 2x³, 6, 3x²y, etc.
Binomial: If the number of terms is two then the expression is known as binomial. Examples- 4x³-3, 7+2y², etc.
Trinomial: If the number of terms is three then the expression is known as trinomial. Examples- 3x²-x+6, 8x+7x³-3, etc.
Types of polynomials based on degree:-
We are classifying the polynomials according to the degree or higher power of the variable.
Linear polynomial: A polynomial of degree 1 is called a linear polynomial. Examples of linear polynomials are 4x-7, 9y+2, etc.
Quadratic polynomial: If the degree of a polynomial is two then it is known as a quadratic polynomial. Example:- 2x²-4x+2, 5y+2y²-10, etc.
Cubic polynomial: The degree of a polynomial is three then it is known as a cubic polynomial. Example:- x²-5x³+7x+8, y³+4y²-6y-6, etc.
What are coefficients in polynomials?
The coefficients are the multiples of variables and the constant terms are the numbers without variables in a polynomial.
Let's understand it.
The general form of a quadratic polynomial is ax²+bx+c=0
Where
x = variable
a = coefficient of x²
b = coefficient of x
c = constant term
but how to find the coefficient of a polynomial? To find the value of the coefficient we compare the given polynomial with the polynomial standard form. Let us take an example:
On comparing 2x²-4x+2=0 with ax²+bx+c=0 we get a=2, b=-4, and c=2.
What is the zero of a polynomial?
Zeroes are values of the variable present in the polynomial. In some cases, zeroes are denoted by α & β of quadratic polynomials.
How to verify the relationship between zeroes & the coefficients?
To verify the relationship between zeroes & the coefficients of the quadratic polynomial we need to find the zeroes and coefficients of the given polynomial and then put the values in the α+β=-b/a, αβ=c/a.
Where α and β are the zeroes of the polynomial and a, b, and c are the coefficients of a polynomial.
Now we take an example to verify the relationship.
Find the zeroes of quadratic polynomial x²+7x+10 & verify the relationship between the zeroes & coefficients.
First, we find the zeroes of the given polynomial x²+7x+10
- x²+7x+10=0
- x²+5x+2x+10=0
- x(x+5)+2(x+5)=0
- (x+5)(x+2)=0
- x+5=0 , x+2=0
- x=-5, x=-2
- α=-5, β=-2
Now, we compare the given quadratic polynomial with the general form to find the coefficients.
ax²+bx+c=0
x²+7x+10=0
On comparing we get a=1, b=7 & c=10.
Verifying the Relationship between zeroes & coefficients:-
Sum of zeroes: α+β = -b/a
Product of zeroes: αβ =c/a
Now put all the above values in the formula-
- α+β = -b/a
- (-5)+(-2) = -7/1
- -5-2 = -7
- -7 = -7
- αβ =c/a
- (-5)×(-2) = 10/1
- 10 = 10
In this way, we verify the relationship between the zeroes & the coefficients of quadratic polynomials.
