### POLYNOMIAL

It is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is defined as an expression which contains two or more algebraic terms. Polynomials are composed of:Constants. Example: 1, 2, 3,1/2, -20 etc.
Variables. Example: g, h, x, y, xy etc.
Exponents: Example: 5 in x2 etc.

For example:- p(x) = 5x3 – 2y2 + 3z – 10
In above polynomial:
Coefficient : 5, 2, 3
Exponent : 3, 2
Variable: x, y, z
Terms: 5x3 , 2y2 , 3z , 10
Contant: 10

### Notation

The polynomial function is denoted by P(a) where a represents the variable. For example,

P(a) = a2 – 5a + 15

### Standard Form

The standard form of writing a polynomial equation is to put the highest degree first then, at last, the constant term.
For example:- 5x4 + 6x3 – 72 + 10x + 6

### Degree

The degree of a polynomial with only one variable is the largest exponent of that variable.

For example:- 4y3 – y – 3 The Degree is 3 (the largest exponent of y)

 Polynomial Degree Example No of Terms Type of Polynomial Constant or Zero Polynomial 0 4 1 MONOMIAL Linear Polynomial 1 4x + 5 2 BINOMIAL Quadratic Polynomial 2 5x2 + 4x + 5 3 TRINOMIAL Cubic Polynomial 3 2x3 1 MONOMIAL Quartic Polynomial 4 7x4 + 7 2 BINOMIAL

### Types of Polynomials

The three types of polynomials are:

#### Monomial

In mathematics, a monomial is a polynomial with just one term.

For example:- 3x, 4xy is monomial.

#### Binomial

In algebra, a binomial is a polynomial, which is the sum of two monomials.

For example:-
2x+5 is a Binomial.

#### Trinomial

In element algebra, a trinomial is a polynomial consisting of three terms or monomials.

### Zeroes of Polynomials

Consider the polynomial p(x) = 5x3-2x2+3x-2.If we replace x by 1 everywhere in p(x), we get
p(1) = 5×(1)3 – 2×(1)2 + 3×(1) – 2
= 5 – 2 + 3 – 2
= 4
So, we say that the value of p(x) at x = 1 is 4.
Similarly,
p(0) = 5(0)3 – 2(0)2 + 3(0) – 2
= – 2.
For example:- 3x + 5y + 7z is a Trinomial.

### Properties of Polynomials

1. If P and Q are polynomials then

Degree(P ± Q) ≤ Degree(P or Q)

Degree(P × Q) = Degree(P) + Degree(Q)

2. If a polynomial P is divisible by a polynomial Q, then every zero of Q is also a zero of P.
3. If a polynomial P is divisible by two coprime polynomials Q and R, then it is divisible by (Q • R).
4. Constant monomials always have a degree of 0.

### Operations of Polynomials

There are four main polynomial operations which are:

2. Subtraction of Polynomials
3. Multiplication of Polynomials
4. Division of Polynomials

METHOD 1:-
Line up like terms. Then add the coefficients.
P = 3x + 7
Q = 2x + 3
P + Q = (3x + 7) + (2x + 3)
= 3x + 2x + 7 + 3
= 5x + 10

METHOD 2:-
Group like terms. Then add the coefficients.
4x2 + 6x + 7 + 2x2 – 9x + 1
= (4x2 + 2x2) + (6x – 9x) + (7 + 1)
= 6x2 – 3x + 8.

The sum of two polynomials is also a polynomial.

#### Subtraction of Polynomials

METHOD 1:-
Line up like terms, change the sign of the second polynomials , then add.
For example:

P = 4x – 7
Q = 2x + 3
P – Q = (4x – 7) – (2x + 3)
= 4x – 7 – 2x – 3
= 4x -2x – 7 – 3
= 2x-10

METHOD 2:-
Simplify :
(5x2 – 3x) – (-82 + 11)
Write the opposite of each term:
5x2 – 3x + 8x2 -11
Group like terms:
(5x2 + 8x2) + (3x + 0) + (-11 + 0)
= 13x2 + 3x – 11.
The difference of two polynomials is also a polynomials.

#### Multiplication of Polynomials

We can multiply two or more than polynomial
For example:- (9y − 3z) × (5y + 5)
= 9y x (5y + 5) – 3z x (5y + 5)
= (9y x 5y + 9y x 5) – (3z x 5y + 3z x 5)
= (45y2 + 45y) – (15zx + 15z)
= 45y2 + 45y – 15zx – 15z)

#### Division of Polynomials

The methods used for finding division of polynomials are :

1- Long division method
2- Factor theorem

##### Long division method

As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient.

Dividend = (Divisor × Quotient) + Reminder

##### Factorisation

Factor theorem:
If p(x) is a polynomial of degree n>1 and a is any real number,
Then,
i) x – a is a factor of p(x), if p(a) = 0,

And

ii) p(a) = 0, if x – a is a factor of p(x).

### Uses of Polynomials

1- Polynomial 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.

2- They are used to define “polynomial functions “, which apper in settings ranging from basic chemistry and physics to economics and social sciences.

3- They are used in calculus and numerical analysis to approximate other functions.