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
Leading Coefficient: 5
Exponent : 3, 2
Variable: x, y, z
Terms: 5x3 , 2y2 , 3z , 10
Contant: 10

Polynomial

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:

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

Addition 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.

I hope, this article will help you a lot to understand the Polynomial. If you still have any doubts and problems with any topic of mathematics you can ask your problem in the Ask Question section. You will get a reply shortly.

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