Class 10 | class 10 Maths | class 10 chapter 2 Maths | polynomial | zeros of polynomial | ncert maths class 10

Chapter 2

What is polynomial ?

"An expression that contains different power of same variable in algrebaic form " .
Example : P(x) = 2x + 5 , P(y) = y² + 5y + 6

What is the Degree Of polynomial ?

" Degree of polynomial is the highest power of variable in given polynomial. "
For Example : P(x) = 2x + 5 has one as the highest power of x , so degree of polynomial is 1
similarly P(y) = y² + 5y + 6 is 2 as the highest power of y , degree polynomial.
Linear Polynomial : Polynomial having degree '1'
Quadratic Polynomial : Polynomial having degree '2'
Cubic Polynomial : Polynomial having degree '3'

Zeroes of a polynomial :

A real number K is said to be zero of polynomial , If P(K) = 0
Example : P(x) = x + 5
For x = -5 ⇒ P(-5) = 0
So , -5 is the zero of the polynomial.

Geometrical meaning Of zeroes :

If we plot or draw any polynomial on x-y plane, and the curve cuts the x-axis at some point , then these points will be known as zeroes of the polynomial.

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Relationship between zeroes and coefficient of a polynomial.

Let's the polynomial p(x) = ax² + bx + c
This polynomial will give 2 zeroes. let the zeroes are α and β
Sum of Zeroes (α + β ) = -(coefficient of x)/(coefficient of x²) = -b/a

Sum of Zeroes (α + β ) = -b/a

Product of zeroes (α . β ) = (constant term) /(coefficient of x²) = c/a

Product of zeroes (α . β ) = c/a

Example : Find the zeros f the polynomial x² + 7x + 10 , and verify the relationship between the zeroes and the coefficients.
solution :
x² + 7x + 10 = x² + 5x + 2x + 10 = (x+5)(x+2)
the Zeroes of Given Polynomial :
x+ 5 = 0 , x = - 5
x+2 = 0 , x= -2
The zeroes are -5 and -2
Verification : Sum of zeroes = -5 -2 = -7
Sum of Zeroes (α + β ) = -b/a = -7/1 = -7
So , Sum of Zeroes (α + β ) = -(coefficient of x)/(coefficient of x²) = -b/a = -7
Product of zeroes = -5*-2 = 10
Product of zeroes (α . β ) = c/a = 10/1 = 10
So , Product of zeroes (α . β ) = (constant term) /(coefficient of x²) = c/a = 10 .
So , it is verified.
Question no. 2
Find the quadratic Polynomial , if sum and product of zeroes are 4 and 1.

Formula : x² - (sum of zeroes)x + Product of zeroes

Sum of zeroes = 4
product of zeroes = 1
Polynomial :
⇒ x² - (sum of zeroes)x + Product of zeroes
⇒ x² - (4)x + 1
⇒ x² - 4x + 1

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