Polynomial Class 10 Notes for board 2021-22[PDF Included]

Polynomial: Those algebraic expressions with integral Exponents and Real Coeffients are called polynomial.

For Example :

f(x)=3x-2 is polynomial [Reason: Power of variable x is 1]

But 3 x½-2 is not polynomial  [Reason: Power of variable x is fractional]

Other Examples …..

f(x)=3 y²-2 y+4 is polynomial [Reason: Power of all variables y is positive integers i.e 1,2]

But f(x) =3 y²-2 y½+4 is not polynomial  [Reason: Power of variables are fractional]

Other Expressions which are also not polynomial : 1/x²-2x+5 , 2x³-3/x+4

Degree of polynomial

Degree of a polynomial: The highest power of the variables of a polynomial is called the Degree of a polynomial.

For Example (older ones):

(i)p(x)=3x-2 ,The highest power of variable x is 1.So, Degree of given Polynomial is 1 .

(ii)p(x)=3y²-2y+4 ,The highest power of variable y is 2.So,Degree of Polynomial is 2 .

Practice this Worksheet on Degree of Polynomial to understand this concept Clearly .

Types of polynomial

Polynomial can be classified on the basis

(i) Degree of polynomial

(ii) Number of terms

Degree of polynomial

Based on the Degree of Polynomial,Polynomials are divided into 5 categories

(i) Constant polynomial: A polynomial of degree 0 is called a constant polynomial.

Example:  f(x)=7 is constant polynomial because the power of x is 0

BUT

Example: f(x)=0 is zero polynomial

Zero polynomial: The polynomial whose all coefficients are zero is called Zero polynomial

Example : 0x³+0x²+0x

Note – The degree of zero polynomial is not defined because no matter how much you increase the power of x, coefficients 0 will make its value always 0 i.e f(x)=0

(ii) Linear polynomial: A polynomial of degree 1 is called a linear polynomial.

Example:  p(x)=4x-3 is linear polynomial because the degree of the polynomial is 1.

The general form of a linear polynomial is f(x)=ax+b

(iii) Quadratic polynomial:A polynomial of the degree of 2 is called a quadratic polynomial.

Example : f(x)=2x²+3x+5 is quadratic polynomial because the degree of the polynomial is 2.

The general form of a quadratic polynomial is ax²+bx+c where a≠0

(iii) Cubic polynomial: A polynomial of degree 3 is called a cubic polynomial.

Example: f(x)=2y³+5y-7 is cubic polynomial because the degree of the polynomial is 3.

The general form of a cubic polynomial is ax³+bx²+cx+d where a≠0

Biquadratic polynomial: A polynomial of the degree of 4 is called a biquadratic polynomial.

Example: f(x)=2y^4+3y³+4y²+1/2y+5 is biquadratic polynomial .

The general form of a biquadratic polynomial is: ax^4+bx³+cx²+dx+e.

Value of a polynomial: If f(x) is polynomial and α is a real number then, f(α) is called the value of the polynomial.

Example : Let f(x)=3x-5 is a Polynomial and Substitute x=1

Let x=1 then,

3(1)-5 =3-5 =-2

Hence ,-2 is the value of Polynomial at x=1 .

Zero of polynomial : If f(x)is polynomial and α is real number f(α) =0 then, α is called Zero of polynomial .

Consider above given example f(x)=3x-5 ,let α is zero of polynomial f(x)

∴  f(α)=0

3(α)-5=0

3α-5=0

α=5/3

So,zero of polynomial f(x) is 5/3.

No. of zeros of polynomial =Degree of the polynomial.

Relationship between Zeros and the coefficient of a polynomial

Linear polynomial

Let f(x)=3x-5 is polynomial and α is a zero of the polynomial .

On comparing with general form of linear polynomial f(x) =ax+b

ax+b=3x-5

a=3 , b=-5

Then, α=5/3 which is equal to -b/a

∴ α is given by -b/a

Consider a quadratic polynomial f(x) =x²-7x+10 .Find its factor by spillting middle term

f(x)=x²-7x +10

=x²-5x-2x +10

=x(x-5)-2(x-5)

=(x-5)(x-2)

For zeroes ,

(x-5)(x-2) =0

∴ (x-5)=0 or (x-2)=0

x=5 or x=2

Sum of zeroes =5+2 =7 =-\frac{(coefficient of x)}{(coefficient of x²)}

Product of zeroes =5×2=10 ={Constant term}{coefficient of x²}

Proof : If α and β are zeroes of a quadratic polynomial f(x)=ax²+bx+c .By factor theorem (x-α) and (x-β) are factors of f(x).

f(x)=k(x-α)(x-β), where k is any non zero real number

=k{x²-(α+β)+αβ}

We know ,General Form of Quadratic Equations ax²+bx+c

ax²+bx+c =kx²-k(α+β)+kαβ

On comparing both sides, we get

a=k ,b=-k(α+β) c=kαβ

Now

-b/a=-{-k(α+β)}/k=(α+β) which is equal to sum of zeroes.

c/a =kαβ/k=αβ which is equal to the product of zeroes.

∴ α+β =\frac{(coefficient of x)}{(coefficient of x²)}

αβ= {Constant term}{coefficient of x²}

Note: If α and β are zeroes of a polynomial then,

The polynomial is given by f(x)=k{x²-(α+β) x+αβ}

or

f(x)=k{x²-(sum of zeroes )x+product of zeroes }

Example: Let we have two zeroes of a polynomial i.e 2,5

the sum of zeroes =2+5=7

product of zeroes =2.5=10

since,f(x)=k{x²-(sum of zeroes )+product of zeroes }

f(x)=k{x²-7x+10}, where k is any non zero real number

Later we will understand with an example of why the value of k is not fixed

Cubic polynomial

Let α,β and γ are zeroes of polynomial f(x)=ax³+bx²+cx+d where a≠0

By factor theorem (x-α),(x-β) and (x-γ) are factors of f(x).

f(x)=k(x-α)(x-β) (x-γ)

f(x)=k{x³-(α+β+γ)x²+(αβ+βγ+γα)x-αβγ}

On comparing with the general form

ax³+bx²+cx+d=k{x³-(α+β+γ)x²+(αβ+βγ+γα)x-αβγ}

ax³+bx²+cx+d=kx³-k(α+β+γ)x²+k(αβ+βγ+γα)x-kαβγ

On comparing on both side we get,

a=k, b=-k(α+β+γ) , c=k(αβ+βγ+γα and d=-kαβγ

since b=-k(α+β+γ) ( a=k)

b=-a(α+β+γ)

(α+β+γ)=-b/a

similarly

αβ+βγ+γα=c/a

αβγ =-d/a

Note: If α,β, and γ are zeroes of cubic polynomial f(x) then,it is given by

f(x)=k{x³-(α+β+γ)x²+(αβ+βγ+γα)x-αβγ} where k is any constant

Let’s understand why the value of k is not fixed

Q. Find the no. of polynomial whose zeroes are 2 and 5.

sol. Let α=2 and β=5

we know, α+β =-b/a

αβ =c/a

since ,  f(x)=k{x²-(sum of zeroes )x+product of zeroes }

∴ f(x)=k{x²-7x+10}

for zeroes ,f(x)=0

k{x²-7x+10}=0

{x²-7x+10}=0/k

{x²-7x+10}=0

Zeroes of a polynomial are independent on the value of x