POLYNOMIALS
• 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 .
DEGREE OF POLYNOMIAL
• 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 .
TYPES OF POLYNOMIALS
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Types of polynomials are –
1] Constant polynomial
2] Linear polynomial
3] Quadratic polynomial
4] Cubic polynomial
5] Bi-quadratic polynomial
CONSTANT POLYNOMIAL
• CONSTANT POLYNOMIAL – A
polynomial of degree zero is called a
constant polynomial.
• EXAMPLE - F(x) = 7 etc .
• It is also called zero polynomial.
• The degree of the zero polynomial is not
defined .
LINEAR POLYNOMIAL
• 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.
QUADRATIC POLYNOMIAL
•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
CUBIC POLYNOMIALS
• 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 .
BI QUADRATIC POLYNMIAL
• BI – QUADRATIC POLYNOMIAL –
A fourth degree polynomial is called a
biquadratic polynomial .
VALUE OF POLYNOMIAL
• If p(x) is a polynomial in x, and if k is any real
constant, then the real number obtained by
replacing x by k in p(x), is called the value of
p(x) at k, and is denoted by p(k) . For
example , consider the polynomial p(x) = x²
3x 4 . Then, putting x= 2 in the polynomial ,
we get p(2) = 2²  3  2  4 =  4 . The value
 6 obtained by replacing x by 2 in x²  3x  4
at x = 2 . Similarly , p(0) is the value of p(x) at
x = 0 , which is  4 .
ZERO OF A POLYNOMIAL
• 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,
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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 .
HOW TO FIND THE ZERO OF
A LINEAR POLYNOMIAL
• In general, if k is a zero of p(x) = ax  b,
then p(k) = ak  b = 0, k =  b  a . So,
the zero of a linear polynomial ax  b is
 b  a =  ( constant term ) 
coefficient of x . Thus, the zero of a
linear polynomial is related to its
coefficients .
GEOMETRICAL MEANING OF
THE ZEROES OF A POLYNOMIAL
• 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
GEOMETRICAL MEANING OF
LINEAR POLYNOMIAL
• 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 .
QUADRATIC POLYNOMIAL
• 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 .
GEOMETRICAL MEANING OF
CUBIC POLYNOMIAL
• 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
.
RELATIONSHIP BETWEEN
ZEROES OF A POLYNOMIAL
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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
RELATIONSHIP BETWEEN ZERO
AND COEFFICIENT OF A CUBIC
POLYNOMIAL
• In general, if  ,  , Y are the zeroes of a
cubic polynomial ax³  bx²  cx  d , then
• Y =  ba
• =  ( Coefficient of x² )  coefficient of x³
•  Y Y =c  a
• = coefficient of x  coefficient of x³
• Y =  d  a
• =  constant term  coefficient of x³
DIVISION ALGORITHEM FOR
POLYNOMIALS
• 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 .
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