2. 1.INTRODUCTION
2.GEOMETRICAL MEANING
OF ZEROES OF THE
POLYNOMIAL
3.RELATION BETWEEN
ZEROES AND COEFFICIENTS
OF A POLYNOMIAL
4.DIVISION ALGORITHM FOR
POLYNOMIAL
5.SUMMARY
6.QUESTIONS AND EXERCISE
3. The national curriculum framework
such that children's life at school must be
linked to their life outside the school. this
principle marks a de portable use from the
legacy of bookish learning and thus the
students have been given provisions to
preface some project reports on certain
subjects. I express my hearty gratitude to
CBSE for providing such an interesting
and board scope topic for our project. I
am really thankful to our respected
subject teacher Ms.Nivedita Saxena who
helped us in a passive way. I would also
like to thank my parents and my friends
123456789101112
131415161718192
0
4.
5. In mathematics, a polynomial is an
expression of finite length
constructed from variables and
constants, using only the operations
of addition, subtraction,
multiplication, and non-negative,
whole-number exponents. Polynomials
appear in a wide variety of areas of
mathematics and science. For
example, they are used to form
polynomial equations, which encode a
wide range of problems, from
elementary word problems to
complicated problems in the sciences;
they are used to define polynomial
functions, which appear in settings
ranging from basic chemistry and
physics to economics and social
science; they are used in calculus and
numerical analysis to approximate
other functions.
6.
7. Let x be a variable n, be a positive integer
and as, a1,a2,….an be constants (real nos.)
Then, f(x) = anxn+ an-1xn-1+….+a1x+xo
anxn,an-1xn-1,….a1x and ao are known as the
terms of the polynomial.
an,an-1,an-2,….a1 and ao are their
coefficients.
For example:
• p(x) = 3x – 2 is a polynomial in variable x.
• q(x) = 3y2 – 2y + 4 is a polynomial in variable y.
• f(u) = 1/2u3 – 3u2 + 2u – 4 is a polynomial in variable u.
NOTE: 2x2 – 3√x + 5, 1/x2 – 2x +5 , 2x3 – 3/x +4 are not polynomials.
8. The exponent of the highest degree term in a polynomial is
known as its degree.
For example:
f(x) = 3x + ½ is a polynomial in the
variable x of degree 1.
g(y) = 2y2 – 3/2y + 7 is a polynomial
in the variable y of degree 2.
p(x) = 5x3 – 3x2 + x – 1/√2 is a
polynomial in the variable x of degree 3.
q(u) = 9u5 – 2/3u4 + u2 – ½ is a
polynomial in the variable u of degree 5.
9. For example:
f(x) = 7, g(x) = -3/2, h(x) = 2
are constant polynomials.
The degree of constant
polynomials is not defined.
For example:
p(x) = 4x – 3, q(x) =
3y are linear polynomials.
Any linear polynomial is
in the form ax + b,
where a, b are real nos.
and a ≠ 0.
It may be a monomial or a binomial. F(x) = 2x – 3
is binomial whereas g (x) = 7x is monomial.
10. A polynomial of degree two is
called a quadratic polynomial.
f(x) = √3x2 – 4/3x + ½, q(w) =
2/3w2 + 4 are quadratic
polynomials with real
coefficients.
Any quadratic is always in the
form f(x) = ax2 + bx +c where
a,b,c are real nos. and a ≠ 0.
A polynomial of degree three
is called a cubic polynomial.
f(x) = 9/5x3 – 2x2 + 7/3x _1/5
is a cubic polynomial in
variable x.
Any cubic polynomial is
always in the form f(x = ax3 +
bx2 +cx + d where a,b,c,d are
real nos.
11. A real no. x is a zero of the
polynomial f(x),is f(x) = 0
Finding a zero of the polynomial
means solving polynomial
equation f(x) = 0.
If f(x) is a polynomial and y is
any real no. then real no.
obtained by replacing x by y in
f(x) is called the value of f(x) at x
= y and is denoted by f(x).
Value of f(x) at x = 1
f(x) = 2x2 – 3x – 2
f(1) = 2(1)2 – 3 x 1 – 2
= 2 – 3 – 2
= -3
Zero of the polynomial
f(x) = x2 + 7x +12
f(x) = 0
x2 + 7x + 12 = 0
(x + 4) (x + 3) = 0
x + 4 = 0 or, x + 3 = 0
x = -4 , -3
17. ☻ A+B =- coefficient of x
Coefficient of x2
= - b
a
☻ AB = constant term
Coefficient of x2
= c
a
18. A+ B + C = -Coefficient of x2 = -b
Coefficient of x3 a
AB + BC + CA = Coefficient of x = c
Coefficient of x3 a
ABC = - Constant term = d
Coefficient of x3 a
19.
20.
21. If f(x) and g(x) are
any two polynomials
with g(x) ≠ 0,then we
can always find
polynomials q(x), and
r(x) such that :
F(x) = q(x) g(x) + r(x),
Where r(x) = 0 or
degree r(x) < degree
g(x)
ON VERYFYING THE
DIVISION ALGORITHM
FOR POLYNOMIALS.
ON FINDING THE
QUOTIENT AND REMAINDER
USING DIVISION
ALGORITHM.
ON CHECKING WHETHER A
GIVEN POLYNOMIAL IS A
FACTOR OF THE OTHER
POLYNIMIAL BY APPLYING
THEDIVISION ALGORITHM
ON FINDING THE REMAINING
ZEROES OF A POLYNOMIAL
WHEN SOME OF ITS ZEROES
ARE GIVEN.
