Polynomial for class 10 for board 2020-21

Polynomial: Those algebraic expressions in which variables power are positive integers and coefficients are real numbers, that expressions are called a polynomial.

Example – (i)f(x)=3 x-2 is polynomial {power of variable x is 1 }

But 3 x½-2 is not polynomial {power of variable x is fractional }

(ii) f(x)=3 y²-2 y+4 is polynomial {power of all variables y is positive integers i.e 1,2 }

But f(x) =3 y²-2 y½+4 is not polynomial{ power of a variable is fractional }

Other Expressions which are also not polynomial

(i)1/x²-2x+5 (ii) 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.

consider previous examples

(i)p(x)=3x-2 The highest power of variable x is 1.

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

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.polynomial is divided into

(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 it value completely 0 i.e f(x)=0

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

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

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

Example f(x)=3x-5

Let x=1 then,

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

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.

The no of zeroes polynomial =Degree of the polynomial.

Relationship between zero and the coefficient of a polynomial

Linear polynomial

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

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

a=3 , b=-5

then, α=5/3 which is equal to -b/a

∴ α is given by -b/a

Quadratic polynomial

consider a quadratic polynomial f(x) =x²-7x+10 .find the factor by spiltting 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 =-(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).

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

=k{x²-(α+β)+αβ}

 

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 sum of zeroes

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

∴ α+β =-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

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