CLASS X MATHS Polynomials

POLYNOMIALS

1. POLYNOMIALSPOLYNOMIALS

2. INTRODUCTION :-
•POLYNOMIALS -
In mathematics, a polynomial is an
expression of finite length constructed
from variables and constants with non-
negative, whole-number exponents.
EG : p (x) = 2x
2
+ x + 3
p (y) = 6y
3
-3y + 5

3. TERMS IN A POLYNOMIAL
•Monomials – polynomials with one term
e.g.:- 4x
3
, 7 (constant term)
Binomials - polynomials with two terms
e.g.:- 3x
6
 + 13x
5
 
•Trinomials - polynomials with three terms
e.g.:- 7x
5
- 9 x
2
+1
While writing a polynomial always the term with
highest exponent is written first . And the other terms
are written in descending order according to their
powers.
EG :- p(x) = 17x
9
+ 4x
7
- 3x
4
+ 2x
2
– 10 .

4. DEGREE OF A POLYNOMIAL
Degree of a polynomial is the highest among the exponents of all its terms.
This highest degree of the polynomial is also called as the order of the
polynomial .
Examples
The degree of the polynomial − 8x
7
+ 4x
3
− 9x + 5 is 7.
The degree of the polynomial 5x
6
+ x
4
− 2x
3
+ 9 is 6.
 Linear polynomial – polynomial with degree one
e.g.:- 2x - 3 , 3z + 4 etc.
 Quadratic polynomial - polynomial with degree two
e.g.:- y
2
– 2 , 4z
2
+
1
/
7 etc.
 Cubic polynomial - polynomial with degree three
e.g.:- x
3
+ 4x
2
– 5 etc.

5. ZEROES OF A POLYNOMIAL
•ZERO OF A POLYNOMIAL – It is that value of the variable at which the
value of the polynomial becomes zero.
The zero of the polynomial is defined as any real value of x, for which the
value of the polynomial becomes zero.
A real number k is a zero of a polynomial p(x) , if p(k) = 0 .
EXAMPLE :-
Q. Find the zero of the polynomial p(x)=2x +3 .
Ans. On equating the polynomial to zero , p(x)=2x +3 = 0
2x = -3
x = -3/2
For checking , we replace x by -3/2 in polynomial p(x)=2x +3
p(-3/2)= 2 (-3/2)+3 = 0
= -3+3 = 0
0=0
Therefore , the zero of the polynomial p(x)=2x +3 is x = -3/2

6. NUMBER OF ZEROES OF A POLYNOMIAL
The graph of a linear polynomial
intersects the x-axis at a maximum
of one point. Therefore, a linear
polynomial has a maximum of
one zero.
The graph of a quadratic
polynomial intersects the x-axis at
a maximum of two points.
Therefore, a quadratic polynomial
can have a maximum of two
zeroes.
The graph of a cubic polynomial
intersects the x-axis at maximum
of three points. A cubic
polynomial has a maximum of
three zeroes.

7. GEOMETRICAL MEANING OF ZEROES OF A POLYNOMIAL
Geometrically zeroes of the polynomial are the points where the graph of the polynomial intersects
x-axis .
If the graph of a polynomial p(x) intersects the x -axis at (k,0), then k is the zero of the polynomial.
The equation ax
2
+ bx + c can have three cases for the graphs.

8. Case I :-
Here, the graph cuts x-axis at two distinct
points A and A′.

9. Case (ii):-

Here, the graph cuts the x-axis at exactly one point

10. Case (iii):-
Here, the graph is either completely above the x-axis or
completely below the x-axis.

11. RELATIONSHIP BETWEEN ZEROES
AND COEFFICIENTS OF A LINEAR
POLYNOMIAL
•The general form of linear polynomial is p(x)=ax+ b where
a≠0 . If its zero is α, then,
EXAMPLE
Q .Find the zero of the polynomial p(x) = 3x + 4
Ans. We know, for a linear polynomial , the zero α = -b/a .
Here , b = 4 and a = 3.
Therefore , -b/a = -4/3.
Hence the zero of the given polynomial p(x) = 3x + 4 is -4/3.

12. RELATIONSHIP BETWEEN ZEROES AND
COEFFICIENTS OF A QUADRATIC POLYNOMIAL
• General form of quadratic polynomial is ax
2
+ bx +c where
a≠0.  Let the two zeroes of quadratic polynomial p(x)= ax
2
+ bx +c
be α and β . Then,

13. Q .Find the sum and product of the zeroes of the polynomial
p(x) = 4x
2
+ 3x +2.
Ans. We know, for a quadratic polynomial ,
α+β = -b/a .
αβ = c/a
Here , a = 4 , b = 3 and c = 2
Therefore , -b/a = -3/4 and
c/a = 2/4 = 1/2
Hence the sum of zeroes of p(x) = 4x
2
+ 3x +2 is -3/4 and
the product of zeroes of p(x) = 4x
2
+ 3x +2 is 1/2
EXAMPLE

14. RELATIONSHIP BETWEEN ZEROES
AND COEFFICIENTS OF A CUBIC
POLYNOMIAL
•General form of cubic polynomial is ax
3
+ bx
2
+ cx + d
where a≠0.  Then,

15. DIVISION ALGORITHM FOR POLYNOMIALS
Let f(x), g(x), q(x) and r(x) be polynomials , then the division
algorithm for polynomials states that
“If f(x) and g(x) are two polynomials such that degree of f(x) is
greater that degree of g(x) where g(x) ≠ 0, then there exists two
unique polynomials q(x) and r(x) such that f(x) = g(x) x q(x) + r(x)
where r(x) = 0 or degree of r(x) less than degree of g(x)”.
= f(x) = g(x) x q(x) + r(x)
Dividend = divisor x quotient + remainder

16. EXAMPLE
Consider the cubic polynomial x
3
- 3x
2
- x + 3.

If one of its zeroes is 1, then x - 1 is a factor of x
3
- 3x
2
- x + 3.
So, you can divide x
3
- 3x
2
- x + 3 by x – 1 and you would get
x
2
- 2x – 3 as the quotient .
Next, you could get the factors of x
2
- 2x - 3, by splitting the middle
term, as : (x + 1)(x - 3).
This would give you:
x
3
- 3x
2
- x + 3 = (x - 1)(x
2
- 2x - 3)
= (x - 1) (x + 1) (x - 3).
Now on equating each of these factors to zero , we will get 1, -1, 3.
So, all the three zeroes of the cubic polynomial are now known to you
as 1, -1, 3.