About Polynomials

2. Presented by:-
N.Gayathri
x

4. CONTENT
 Polynomial
 Degree of Polynomial
 Types of Polynomial
 Value of Polynomial
 Zero of a polynomial
 Geometrical meaning of the Zero of a polynomial
 Relationship between Zeroes and coefficients of a
polynomial
 Division algorithm for polynomial

5. polynomial
Algebraic expression
in the form of
a0 + a1x1 + a2x2 + ......
+ an.xn
where a0, a1, a2 .....
an are real numbers, n
is a non-negative
integer and an ≠ 0 is
called a Polynomial
degree n.

6. Degree of polynomial
The higest
power of x in a
polynomial p(x)
is called degree
of polynomial.

7. Types of polynomial
Constant polynomial
Linear polynomial
Quadratic polynomial
Cubic polynomial
Bi-quadratic polynomial

8. Constant polynomial
A polynomial of degree zero is called
a constant polynomial and it is of the
form p(x) = K
Example:-
F(x) = 7

9. Linear polynomial
A polynomial of degree one is
called a Linear polynomial and it is
of the form p(x) = ax + b, where a,b
are real numbers and a ≠ 0
Example:-
5x + 7

10. Quadratic polynomial
A polynomial of degree two is called a
Quadratic polynomial and it is of the form
p(x) = ax2​ + bx + c, where a,b,c are real
numbers and a ≠ 0
Example:-
10x2 + 3x + 7

11. Cubic polynomial
A polynomial of degree three is called
a Cubic polynomial and it is of the
form
p(x) = ax3 + bx2​​ + cx + d, where
a,b,c,d are real numbers and a ≠ 0
Example:-
12x3 + 2x2​​ + 4x + 1

12. Bi-Quadratic polynomial
A polynomial of degree four is called a Bi-
Quadratic polynomial and it is of the form
p(x) = ax4 + bx3 + cx2​​ + dx + e where
a,b,c,d,e are real numbers and a ≠ 0
Example:-
3x4 + 5x3 + 10x2 + 2x + 7

13. Value of a polynomial
If p(x) is a polynomial in the form
of: a0 + a1x1 + a2x2 + ...... + an.xn
with a ≠ 0, then value of the
polynomial p(x) for a real value at
x = a will be given by
p(a) = a0 + a1 * (a) + a2 * (a)2 +
a3 * (a)3 + .... + an * (a)n

14. Zero of a polynomial
A number ‘a’ is
said to be the
zero of a
polynomial p(x),
if on replacing
each x in the
polynomial by ‘a’
the value of a

15. Geometrical meaning of
the zeroes of a
polynomial

16. the number
of curves
tells the
degree.
the number
of time it cut
the x-axis

17. P(x) does not
meet x-axis any
where it means
the given
polynomial
does not has
any real zeros.
y
x’ x
y’

18. If graph of any
polynomial p(x)
meet the x-axis
only at one point
then it has only
one zero and
polynomial will be
linear polynomial
y
x ‘ x
y’

19. y
X’ x
y
 If graph of
polynomial p(x)
either cut the x-
axis at two
points then it has
two real zeros
and polynomial
is said to
quadratic
polynomial

20. y
x’ x
y’
 If graph of
polynomial p(x)
intersects x-axis at
three distinct points
or intersects at one
point also touches x-
axis then the
polynomial has three
real zeros and is said
to be cubic
polynomial

21. y
x’ x
y’
A polynomial
is said to has
four real zeros
if the graph of
the
polynomial
p(x) intersect
x-axis at four
point

22. Relationship between
Zeroes and coefficients of
a polynomial

23. Let A and B be the zeros of the
polynomial ax2 + bx + c
then
 sum of zeroes(α+β)= - Coefficient of x - b

 Coefficient of x a
 Product of zeroes (α+β)= - Coefficient of x -
 Coefficient of x

24. Division algorithm
for polynomial

25. If p(x) and g(x) are any
two polynomials with
g(x) 0, then we can find
polynomials q(x) and
r(x)
such that
p(x) = q(x) * g(x) + r(x)
= 0

26. Divide 2x2 + 3x – 2 by x+2
2x - 1
x + 2 2x2 + 3x - 2
2x2 + 4x
-x - 2
-x - 2
0