1.
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.
2.
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.
3.
•A polynomial of degree 1 is called a Linear
Polynomial. Its general form is ax+b where a is not
equal to 0
•A polynomial of degree 2 is called a Quadratic
Polynomial. Its general form is ax3+bx2+cx, where a
is not equal to zero
•A polynomial of degree 3 is called a Cubic
Polynomial. Its general form is ax3+bx2+cx+d,
where a is not equal to zero.
•A polynomial of degree zero is called a Constant
Polynomial
4.
LINEAR POLYNOMIAL
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.
QUADRATIC POLYNOMIAL
For example:
f(x) = √3x2 – 4/3x + ½, q(w) = 2/3w2 + 4
are quadratic polynomials with real
coefficients.
5.
For example:
If(x) = 7, g(x) = -3/2, h(x) = 2
are constant polynomials.
CUBIC POLYNOMIAL
CONSTANT POLYNOMIAL
For example:
f(x) = 9/5x3 – 2x2 + 7/3x _1/5
is a cubic polynomial in variable x.
6.
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 POLYNOMIAL
ZEROES OF POLYNOMIAL
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.
11.
α + β = - coefficient of x
Coefficient of x2
= - b
a
αβ = constant term
Coefficient of x2
= c
a
12.
α + β + γ = -Coefficient of x2 = -b
Coefficient of x3 a
αβ + βγ + γα = Coefficient of x =
c
Coefficient of x3 a
αβγ = - Constant term = d
Coefficient of x3 a
14.
•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.
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