1) The document defines different types of polynomials including linear, quadratic, and cubic polynomials. It gives examples of each type. 2) Key information about polynomials includes that the degree refers to the highest power of the variable, and that a polynomial's zeros are the values where it equals 0. 3) Properties of polynomial zeros are discussed, such as that a linear polynomial has 1 zero, a quadratic polynomial has up to 2 zeros, and a cubic polynomial has up to 3 zeros. Relations between coefficients and zeros are also presented.

1. Name - Ankit Goel
Class – X D
Roll No. - 18
Submitted To:
Ms. Neeru

2. POLYNOMIAL – A polynomial in one
variable X is an algebraic expression in X of
the form
NOT A POLYNOMIAL – The expression
like 1x  1,x+2 etc are not polynomials .

3. Degree of polynomial - The highest power of x
in p(x) is called the degree of the polynomial
p(x).
EXAMPLE –
1. F(x) = 3x +½ is a polynomial in the variable
x of degree 1.
2. g(y) = 2y²  ⅜ y +7 is a polynomial in the
variable y of degree 2 .

4. CONSTANT POLYNOMIAL – A polynomial of
degree zero is called a constant polynomial.
EXAMPLE - F(x) = 7 etc .
1. It is also called zero polynomial.
2. The degree of the zero polynomial is not defined
.

5. LINEAR POLYNOMIAL – A polynomial
of degree 1 is called a linear polynomial .
EXAMPLE - 2x3 , 3x +5 etc .
The most general form of a linear
polynomial
is ax  b , a  0 ,a & b are real.

6. QUADRATIC POLYNOMIAL – A
polynomial of degree 2 is called quadratic
polynomial .
EXAMPLE – 2x²  3x  ⅔ , y²  2 etc .
More generally , any quadratic polynomial in x
with real coefficient is of the form ax² + bx + c,
where a, b, c are real numbers and a  0

7. CUBIC POLYNOMIAL – A polynomial of
degree 3 is called a cubic polynomial .
EXAMPLE = 2  x³ , x³, etc .
The most general form of a cubic polynomial
with coefficients as real numbers is ax³  bx² 
cx  d , a ,b ,c ,d are reals .

8. A real number k is said to a zero of a polynomial
p(x), if said to be a zero of a polynomial p(x), if
p(k) = 0 . For example, consider the polynomial
p(x) = x³  3x  4 . Then,
p(1) = (1)²  (3(1)  4 = 0
Also, p(4) = (4)²  (3 4)  4 = 0
Here,  1 and 4 are called the zeroes of the
quadratic polynomial x²  3x  4 .

9. We know that a real number k is a zero of the
polynomial p(x) if p(K) = 0 . But to understand
the importance of finding the zeroes of a
polynomial, first we shall see the geometrical
meaning of –
1) Linear polynomial .
2) Quadratic polynomial
3) Cubic polynomial

10. For a linear polynomial ax  b , a  0, the
graph of y = ax b is a straight line . Which
intersect the x axis and which intersect the x
axis exactly one point ( b  2 , 0 ) .
Therefore the linear polynomial ax  b , a 
0 has exactly one zero .

11. For any quadratic polynomial ax²  bx c, a 
0, the graph of the corresponding equation y =
ax²  bx  c has one of the two shapes either
open upwards or open downward depending on
whether a0 or a0 .these curves are called
parabolas .

12. The zeroes of a cubic polynomial p(x) are the x
coordinates of the points where the graph of y =
p(x) intersect the x – axis . Also , there are at most 3
zeroes for the cubic polynomials . In fact, any
polynomial of degree 3 can have at most three
zeroes .

13. For a quadratic polynomial – In general, if  and  are the
zeroes of a quadratic polynomial p(x) = ax²  bx  c , a  0 ,
then we know that x   and x  are the factors of p(x) .
Therefore ,
ax²  bx  c = k ( x  ) ( x   ) ,
Where k is a constant = k[x²  (  )x ]
= kx²  k(    ) x  k 
Comparing the coefficients of x² , x and constant term on
both the sides .
Therefore , sum of zeroes =  b  a
=  (coefficients of x)  coefficient of x²
Product of zeroes = c  a = constant term  coefficient of x²

14. In general, if  ,  , Y are the zeroes of a cubic
polynomial ax³  bx²  cx  d , then
1. Y =  b  a
=  ( Coefficient of x² )  coefficient of x³
2.  Y Y =c  a
= coefficient of x  coefficient of x³
3. Y =  d  a
=  constant term  coefficient of x³

15. 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)
Where r(x) = 0 or degree of r(x)  degree of g(x) .
This result is taken as division algorithm for
polynomials .