A polynomial
is an algebraic expression of the form a_{n}x^{n} + a_{n1}x^{n1}
+ a_{n2}x^{n2} + …………… + a, where n is a nonnegative integer
and the coefficients an, a_{n1}, a_{n2}, ………, a are real
numbers with a_{n} ≠ 0. It is called a polynomial of degree n in
variable x.
Some examples of polynomial are:

3x + 5

2x^{2} + x – 1

X^{3} + 2x^{2}
+ 5x + 6
Multinomial and polynomial do not mean
the same thing. All polynomials are multinomial, but all multinomial may not be
polynomials. For example: 3x^{3} + 2x^{2} + 6x + 2√x – 9 is a
multinomial but not a polynomial.
A polynomial in a variable x is said to
be in standard form, if it is in simplified form and arranged in ascending or
descending order. A polynomial in variable x is usually denoted by the symbols
f(x), g(x), h(x), etc.
The standard form of a polynomial is in the form of:
f(x)= a_{n}x^{n} +a_{n1}x^{n1} +a_{n2}x^{n2}
+ …………….... +a.
or, g(x)
= a + a_{1}x + a_{2}x^{2} + …………….... + a_{n}x^{n}.
(i) If a_{n} ≠ 0, then n is called the degree of the
polynomial f(x). Degree of polynomial can never be negative.
(ii) a_{n}x^{n}, a_{n1}x^{n1}, a_{n2}x^{n2},
……… , a are called the terms of the polynomial f(x). a is called the constant
term.
(iii) a, a_{1}, a_{2}, ……….., a_{n} are called
the coefficients of the polynomial f(x).
(iv) If a_{n} ≠ 0, then a_{n}x^{n} is called
the leading term and a_{n} is called the leading coefficient of the
polynomial.
(v) If all the coefficients a_{n}, a_{n1}, a_{n2},
………. , a are zero, then f(x) is called zero polynomial. The degree of zero
polynomial is never defined.
(vi) The degree of a polynomial is zero if and only if it is a nonzero
constant polynomial. For example, f(x) = 5 is a zero degree polynomial.
A polynomial of degree one (1) is called linear polynomial. Polynomials of
degree 2, 3 and 4 are respectively called quadratic
polynomial, cubic polynomial and
biquadratic polynomial. For example:

f(x) = ax + b is linear
polynomial.

f(x) = ax^{2} + bx +
c is quadratic polynomial.

f(x) = ax^{3} + bx^{2}
+ cx + d is cubic polynomial.

f(x) = ax^{4} + bx^{3}
+ cx^{2} + dx + e is biquadratic polynomial.
A polynomial, non of whose coefficients
is zero is called a complete polynomial. Otherwise it is an incomplete
polynomial.
If the coefficients of a polynomial are
integers, then the polynomial is said to be the polynomial over integers. f(x)
= 5x^{3} – 3x^{2} + 3x + 2 is a polynomial over integers. If
the coefficients of a polynomial are rational numbers, it is called the
polynomial over rationals. A polynomial is called polynomial over real, if its
coefficients are real numbers.
If a polynomial contains two variables x and y, it
is regarded as a function of x and y.
For example,
f(xy) = x^{2}y
+ 5xy + y^{2}
A polynomial containing three variables x, y and z
is written as follows:
F(xyz)
= x^{2}yz + x^{2}z + xz^{2} + xy^{2}z + 2.
The degree of polynomial in two or more variables is the highest
sum of the powers of the term involved in the polynomial.
For example:

f(xy) = 3xy^{3} – x^{2}
+ 5x – 8 is a polynomial of degree 4.

f(xy) = 5x^{3}y^{4}
is a polynomial of degree 7.
Equal polynomials
Two polynomials are said to be equal if
and only if
(i) their degree are same
(ii) coefficients of corresponding
terms are same.
If f(x) = a_{n}x^{n} + a_{n1}x^{n1}
+ a_{n2}x^{n2} + …………… + a_{0} and g(x) = b_{m}x^{m}
+ b_{m1}x^{m1} + b_{m2}x^{m2} + …………… + b_{0}
are equal polynomials, then n = m, a_{n} = b_{m}, a_{n1}
= b_{mn}, …………… , a_{0} = b_{0}.
If polynomials f(x) and g(x) are equal,
then we write f(x) = g(x).
Addition and subtraction of polynomials
Let f(x) and g(x) be two polynomials
given as follows:
f(x)= a_{0} +a_{1}x + a_{2}x^{2} + …………... + a_{n}x^{n}
g(x)= b_{0} + b_{1}x + b_{2}x^{2} + …………... + b_{n}x^{n}
Then their sum is defined as
f(x) + g(x) = (a_{0} + b_{0})
+ (a_{1} + b_{1})x + (a_{2} + b_{2})x^{2}
+ ………… + (a_{n} + b_{n})x^{n}.
Thus the sum of two polynomials can be
found by grouping the power terms, retaining their signs and adding the
coefficients of like powers.
For the calculation of sum or difference
of polynomials, the polynomials are first kept in standard form. Then, the
coefficients of like terms are added for sum and subtracted for difference.
Example: Let, f(x) = 3x^{2} +
5x + 6 and g(x) = 4x^{2} – 2x + 1, then
 f(x) + g(x) = (3 + 4)x^{2}
+ (5 – 2)x + (6 + 1)
= 7x^{2}
+ 3x + 7
 f(x) – g(x) = (3 – 4)x^{2}
+ (5 + 2)x + (6 – 1)
= x^{2}
+ 7x + 5
Properties of addition of polynomials
1. Closure property: The sum of two polynomials over real numbers is also a
polynomial over real numbers. This is called the closure property of addition. i.e.
polynomials are closed under addition.
2. Commutative property: For any two polynomials f(x) and g(x),
f(x) + g(x) = g(x) + f(x).
This is called the commutative property of addition.
3. Associative property: For any three polynomials f(x), g(x) and h(x)
f(x) + [g(x)+h(x)] = [f(x)+g(x)] + h(x)
This is called the associative property of addition.
4. Additive identity: For any polynomial f(x), there is a polynomial 0 such that
f(x) + 0 = f(x)
Then
0 is called the additive identity.
5. Additive inverse: For any of the polynomial f(x) there is a polynomial –f(x) such that
f(x) + [f(x)] = 0, the additive identity.
Then
–f(x) is called additive inverse of f(x).
You can comment your
questions or problems regarding the polynomials
here.
0 comments: