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 x^{2} etc.
For example:- p(x) = 5x^{3} – 2y^{2} + 3z – 10
In above polynomial:
Coefficient : 5, 2, 3
Leading Coefficient: 5
Exponent : 3, 2
Variable: x, y, z
Terms: 5x^{3} , 2y^{2} , 3z , 10
Contant: 10
Notation
The polynomial function is denoted by P(a) where a represents the variable. For example,
P(a) = a^{2} – 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:- 5x^{4} + 6x^{3} – 7^{2} + 10x + 6
Degree
The degree of a polynomial with only one variable is the largest exponent of that variable.
For example:- 4y^{3} – 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 |
5x^{2} + 4x + 5 |
3 |
TRINOMIAL |
Cubic Polynomial |
3 |
2x^{3} |
1 |
MONOMIAL |
Quartic Polynomial |
4 |
7x^{4} + 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) = 5x^{3}-2x^{2}+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.
4x^{2} + 6x + 7 + 2x^{2} – 9x + 1
= (4x^{2} + 2x^{2}) + (6x – 9x) + (7 + 1)
= 6x^{2} – 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 :
(5x^{2} – 3x) – (-8^{2} + 11)
Write the opposite of each term:
5x^{2} – 3x + 8x^{2} -11
Group like terms:
(5x^{2} + 8x^{2}) + (3x + 0) + (-11 + 0)
= 13x^{2} + 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)
= (45y^{2} + 45y) – (15zx + 15z)
= 45y^{2} + 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.
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