Presentation on polynomials chapter class 10..i hope this helps..!!

2. Polynomials

3. Degree of a polynomial

4. Constant polynomial
 A polynomial of degree 1 is called a constant
polynomial.
 For example, f(x) = 7 , g(x) = -1 etc are constant
polynomial.
Linear Polynomial
 A polynomial of degree 1 is called a linear polynomial.
 Example , f(x) = 2x2+3 , g(x) = 3-x2 etc are linear
polynomials.

5. Quadratic polynomial

6. Bi-quadratic polynomial
 A forth degree polynomial is called a biquadratic polynomial .
For example
 3 x4 – 2x3+3x2-12x+10 is a bi quadratic polynomial in variable
y .
Value of the polynomial
 If f(x)is a polynomial and α is any real number then the real
number obtained by replacing x by α in f(x) , is called the value
of f(x) at x=α and is denoted by f(x).

7. Zero of the polynomial
 A real number α is a zero of the polynomial f(x) , if (α)=0
 Finding a zero of a polynomial f(x) means solving the
polynomial equation f(x)=0. Linear polynomial f(x)=ax+b ,
a≠0 has only one zero α given by
α= -b = - Constant term
a Coefficient of x

8. Graph of a polynomial
 In algebraic and set theoretic language the graph of a
polynomial f(x) is the collection of all points(x,y),where
y=f(x). In geometrical or in grapical language the graph of a
polynomial f(x) is a smooth free hand curve passing through
points (x1,y1) , (x2,y2),(x3,y3) … etc ,where y1, y2,y3
are the values of the polynomial f(x) at x1, x2, x3…
respectively.
 Graph of Linear polynomial
A linear polynomial ax+b=0, a≠0, the graph of y=ax+b is a
straight line which intersects the x-axis at exactly one point ,
namely (-b , 0). Therefore, the linear polynomial ax+b=0 has
exactly one zero

9. Graph of quadratic polynomial
 Any quadratic polynomial ax2+bx+c=0 , a≠0 the graph of
corresponding equation y=ax2+bx+c has one of the two shapes
either open downwards like or open upwards like
depending on whether a>0 or a<0. These curves are called
parabolas.
 Graph of cubic polynomial
The graph of a cubic polynomial cross the x-axis at least once
and almost thrice. Like or .

10. Relationship between the zero and the co-
efficient of polynomial
 Consider the quadratic polynomial f(x)= 6x2-x-2 . By the
method of splitting the middle term , we have
 f(x)= 6x2-x-2 = 6x2-4x+3x-2 = 2x(3x-2)+1(3x-2)
or , f(x)=(3x-2)(2x+1)
f(x)=0
(3x-2)(2x+1)=0
(3x-2)=0 or, 2x+1=0
x=2 or, x= -1
3 2
Hence, the zeroes of 6x2-x-2 are α=2 and β= -1
3 2

11. Co efficient of x
Sum of its zeroes =α+β = 2- 1=1= Co efficient of x2
3 2 6 Constant term
Product of its zeroes=αβ= 2 x -1 = -1 =-2 = Coefficient of x2 .
3 2 3 6
Relationship between the zeroes and coefficients of a
quadratic polynomial
Let α and β be the zeroes of the quadratic polynomial f(x)=a x2+bx+c. By
factor theorem (x-α) and (x-β) are the factors of f(x).
f(x)=k(x-α)(x-β), where k is a constant
ax2+bx+c=k{x2 - (α+ β)x+ αβ}
ax2+bx+c=kx2-k (α+ β)x+kαβ

12.  Comparing the coefficients of x2 , x and constant terms on both sides, we
get
a=k,b-k(α+ β)and c= kαβ
α+ β= -b and αβ = c
a a
Coefficient of x Constant term
α+ β=-Coefficient of x2 And αβ = Coefficient of x2
Hence , Coefficient of x
Sum of the zeroes = -b = Coefficient of x2
a
constant term
Product of zeroes = c =coefficient of x2

13. Relationship between zeroes and Coefficient of a
cubic polynomial
 Let α,β,γ be the zeroes of a cubic polynomial f(x)= ax3+ bx2+cx+d ,
a≠0.Then , by factor theorem , x- α , x- β, and x- γ are factors of f(x).Also ,
f(x),being a cubic polynomial ,cannot have more than three linear
factors.
f(x)= k(x- α)(x- β)(x- γ)
ax3+ bx2+cx+d = k(x- α)(x- β)(x- γ)
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
Comparing the coefficients of x3 , x2 , x and constant terms on both sides
, we get
a=k ,b= -k(α+β+γ) , c =k (αβ+ βγ+ γα)x and d = -k(αβγ)
α+β+ γ = -b/a
αβ+ βγ+ γα = c/a
αβγ = -d/a

14. - b Coefficient of x
sum of the zeroes= a = Coefficient of x3
product of the zeroes taken two at a time = c = Coefficient of x
a Coefficient of x3
Product of the zeroes = -d = -Constant term
a Coefficient of x3

15. Division algorithm for polynomial
Dividend=quotient x divisor +remainder… This is Euclid’s
division lemma.
Division algorithm
If f(x) and g(x) are ant two polynomials with g(x)≠0, then
we can always find polynomials q(x) and r(x) such that
f(x)= g(x) x q(x) + r(x) , where r(x)=0 or
degree r(x)=0 or degree r(x)< degree g(x).
REMARK : If r(x)=0, then polynomial g(x) is a factor of
polynomial f(x).