Polynomial Chapter 2
Different types of polynomials, degree of polynomials, relationship between zeroes and coefficients of quadratic and cubic polynomials, pair of linear equations in two variables, methods of solving pair of linear equations in two variables.
COMMON TOPICS OF IX & X STANDARD
Polynomial:-
An algebraic expression in which the exponent of the variable is a whole number
is called a polynomial. Example : 4x^{4}
- 3x^{3} + 4x^{2} - 5x + 2
All exponents of the variable are whole number, so it is a polynomial
Highest exponent = 4,
Degree = 4
Coefficient of x^{4} = 4, Coefficient of x^{3} = - 3
Coefficient of x^{2} = 4, Coefficient of x = - 5
Constant Term = 2
Types of polynomial On the basis of
terms:-
Monomial:- Polynomial
having one term. Eg. 4x^{2}
Binomial:- Polynomial
having two terms. Eg. 4x^{2} +
6x
Trinomial:- Polynomial
having three terms. Eg. 4x^{2} +
6x + 5
Quadrinomial:-Polynomial having four terms. Eg. 4x^{3}
+ 6x^{2} + 5x + 2
Degree of a Polynomial:
In a polynomial highest exponent of the variable is called its degree.
Example :
Let P(x)= 2x^{4} - 3x^{3}
+ 4x^{2} - 2x + 1 is any polynomial
Highest exponent of the variable = 4
So its degree = 4
Types of polynomial On the basis of degree:-
Constant
Polynomial:-
A polynomial of degree zero is called constant polynomial
For Example:- 3, 5
3 can be written as 3x^{0} ,
Here exponent (power) of x is zero, so its degree is "0"
Linear
Polynomial:-
Polynomial of degree
one is called linear polynomial.
For Example:- P(x)=
ax + b
Quadratic
Polynomial:-
Polynomial of degree two is
called Quadratic polynomial.
For Example:- P(x)= ax^{2}
+ bx + c
Cubic
Polynomial:-
Polynomial of degree three is
called cubic polynomial.
For Example:- P(x) =
ax^{3} + bx^{2} + cx + d
Bi-Quadratic
Polynomial:-
Polynomial of degree four is
called linear polynomial.
For Example:- P(x) = ax^{4} + bx^{3}
+ cx^{2} + dx + e
Zero polynomial:-
A
polynomial with coefficient zero is called zero polynomial.
Zero polynomial can be written as: 0 = 0x^{2} or 0 = 0x^{5} , 0 = 0x^{10}
It means the degree of a zero polynomial is not fixed. So we can say the degree of a zero polynomial is not defined.
Note:- Degree of zero
polynomial is not defined.
Zeroes
of polynomial:-
Solutions of a polynomial are called its zeroes.
Values of x for which the given polynomial
become zero are called the zeroes of the polynomial.
Note:- Number of zeroes of a polynomial is equal to the degree
of that polynomial.
For Example:- A linear polynomial has one zero, Quadratic polynomial has two zeroes, Cubic
polynomial has three zeroes and so on.
Graphically:
No. of zeroes of a polynomial = No. of times the graph of that polynomial intersect the x-axis.
Above graph intersect the x-axis at three points so this polynomial have three zeroes.
Above graph intersect the x-axis at three points so this polynomial have three zeroes.
Graph of a Quadratic Polynomial
Graph of a quadratic polynomial is a parabola, which is either upward parabola or downward parabola.
Upward parabola: Graph of a quadratic polynomial of the type P(x)= ax^{2} + bx + c is called upward parabola. because here the coefficient of x^{2} is positive.
Downward parabola: Graph of a quadratic polynomial of the type P(x)= - ax^{2} + bx + c is called downward parabola. because here the coefficient of x^{2} is negative.
Difference between Quadratic Polynomial and quadratic equations
Quadratic Polynomial | Quadratic Equation |
Quadratic Polynomial can be written as: P(x) = ax^{2} + bx + c | Quadratic Equation can be written as :- ax^{2} + bx + c = 0 |
Solution of quadratic polynomials are called its zeroes. | Solution of quadratic equations are called its roots. |
Remainder Theorem
If any polynomial P(x) is divided by any polynomial x + 5 (say), then P(- 5) becomes the remainder.
Factor Theorem
If any polynomial P(x) is divided by any polynomial x + 5 (say), and P(- 5) becomes = 0, then x + 5 is called the factor of P(x)
SPECIFIC TOPIC FOR CLASS X ONLY
Relationship between zeroes and coefficients of a quadratic polynomial:-
Quadratic polynomial is given by P(x) = ax^{2} + bx + c
If α and β are the zeroes of quadratic polynomial then Division Algorithm:-
Dividend = Divisor x Quotient + Remainder or
p(x) = g(x) x q(x) + r(x)
Operations on Polynomials
Addition of two polynomials
Example:
Add: 2x^{4} + 3x^{2} - 4x + 5 and 3x^{4} - 5x^{3} + 8x – 7
In place of and use ‘+’ sign we get
(2x^{4 }+ 3x^{2} - 4x + 5) + (3x^{4} - 5x^{3} + 8x – 7)
Collect the like terms with same degree
(2x^{4} + 3x^{4}) + (- 5x^{3}) + (3x^{2}) + (- 4x +8x ) + (5 – 7)
5 x^{4} - 5x^{3} + 3x^{2} + 4x + (-2)
5 x^{4} - 5x^{3} + 3x^{2} + 4x - 2
Subtraction of two polynomials
Example: Subtract : 3x^{2} + 4x from 5x^{3} + 2x^{2} – 1
First write the polynomial after from then ‘_’ sign and then first polynomial
(5x^{3} + 2x^{2} – 1) – (3x^{2} + 4x )
Open the brackets and change the sign of the polynomial which is after the negative sign
5x^{3} + 2x^{2} – 1 - 3x^{2} - 4x
Now collect the like terms and apply the operation according to the sign
5x^{3} + (2x^{2} - 3x^{2}) – 4x – 1
5x^{3} + (- x^{2}) – 4x – 1
5x^{3} + - x^{2} – 4x – 1
Multiplication of two polynomials
Example : Multiply (2x + 3)(7x - 4)
2x(7x - 4) + 3(7x - 4)
14 x^{2} – 8x + 21x -12
14 x^{2} + 13x -12
Deleted Topics from the CBSE Syllabus
Relationship between zeroes and coefficients of a Cubic polynomial:-
Cubic polynomial is given by P(x) = ax^{3} + bx^{2} + cx + d = 0
If α, β and 𝜸 are the zeroes of cubic polynomial then
\[Sum \: of\: zeroes\: (\alpha +\beta+\gamma )=\frac{-\left ( coefficient\:of\:x^{2}\right )}{coefficient\: of\: x^{3}}=\frac{-b}{a}\]\[Sum\; of\; Product\; of\; zeroes\; taken\; two \: at\: a\: time\:(\alpha \beta+\beta \gamma +\gamma \alpha )=\frac{coefficient\: of\: x}{coefficient \: of\: x^{3}} =\frac{c}{a}\]\[Product \: of\: zeroes\: (\alpha \beta\gamma )=\frac{constantt\: term}{coefficient \: of\: x^{3}} =\frac{-d}{a}\]
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Polynomials basic concepts-cbse mathematics
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