Polynomials basic concepts-cbse mathematics

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 : 4x⁴ - 3x³ + 4x² - 5x + 2

All exponents of the variable are whole number, so it is a polynomial

Highest exponent = 4,

Degree = 4  

Coefficient of x⁴ = 4,   Coefficient of x³ = - 3

Coefficient of x² = 4,   Coefficient of x = - 5

Constant Term = 2

Types of polynomial On the basis of terms:-

Monomial:- Polynomial having one term. Eg.  4x²

Binomial:-   Polynomial having two terms. Eg.   4x² + 6x

Trinomial:- Polynomial having three terms. Eg.   4x² + 6x + 5

Quadrinomial:-Polynomial having four terms. Eg.   4x³ + 6x² + 5x + 2

Degree of a Polynomial:

In a polynomial highest exponent of the variable is called its degree.
Example :
Let P(x)= 2x⁴ - 3x³ + 4x² - 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 3x⁰ , 

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)= ax² + bx + c

Cubic Polynomial:-
Polynomial of degree three  is called cubic polynomial.   
For Example:- P(x) = ax³ + bx² + cx + d

Bi-Quadratic Polynomial:-
Polynomial of degree four  is called linear polynomial. 
For Example:- P(x) = ax⁴ + bx³ + cx² + dx + e

Zero polynomial:- 
A polynomial with coefficient zero is called zero polynomial.

Zero polynomial can be written as: 0 = 0x² or 0 = 0x⁵ , 0 = 0x¹⁰ 

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)= ax² + bx + c is called upward parabola. because here the coefficient of x² is positive.

Downward parabola: Graph of a quadratic polynomial of the type P(x)= - ax² + bx + c is called downward parabola. because here the coefficient of x² is negative.

Difference between Quadratic Polynomial and quadratic equations

Quadratic Polynomial	Quadratic Equation
Quadratic Polynomial can be written as: P(x) = ax2 + bx + c	Quadratic Equation can be written as :-  ax2 + 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)

Click Here for Algebraic Identities

Operations on Polynomials

Addition of two polynomials

Example:

Add: 2x⁴ + 3x² - 4x + 5 and 3x⁴ - 5x³ + 8x – 7

In place of and  use ‘+’ sign we get

(2x⁴ + 3x² - 4x + 5) + (3x⁴ - 5x³ + 8x – 7)

Collect the like terms with same degree

(2x⁴ + 3x⁴) + (- 5x³) + (3x²) + (- 4x +8x ) + (5 – 7)

5 x⁴ - 5x³ + 3x² + 4x + (-2)

5 x⁴ - 5x³ + 3x² + 4x - 2

Subtraction of two polynomials

Example: Subtract :  3x² + 4x from 5x³ + 2x² – 1

First write the polynomial after from then ‘_’ sign and then first polynomial

(5x³ + 2x² – 1) – (3x² + 4x )

Open the brackets and change the sign of the polynomial which is after the negative sign

5x³ + 2x² – 1 - 3x² - 4x

Now collect the like terms and apply the operation according to the sign

5x³ + (2x² - 3x²) – 4x – 1

5x³ + (- x²) – 4x – 1

5x³ + - x² – 4x – 1

Multiplication of two polynomials

Example : Multiply  (2x + 3)(7x - 4)

2x(7x - 4) + 3(7x - 4)

14 x² – 8x + 21x -12

14 x² + 13x -12

SPECIFIC TOPIC FOR CLASS X ONLY

Relationship between zeroes and coefficients of a quadratic polynomial:-

Quadratic polynomial is given by    P(x) = ax² + bx + c

If α and β are the zeroes of quadratic polynomial then

Relationship between zeroes and coefficients of a Cubic polynomial:-  

Cubic polynomial is given by     P(x) = ax³ + bx² + cx + d = 0

If α, β and 𝜸 are the zeroes of cubic polynomial then

\[Sum \: of\: zeroes\: (\alpha +\beta+\gamma )=\frac{-\left ( coefficient\:of\:x^{2}\right )}{coefficient\: of\: x^{3}}=\frac{-b}{a}\]
\[Sum\; of\; Product\; of\; zeroes\; taken\; two \: at\: a\: time\:(\alpha \beta+\beta \gamma +\gamma \alpha )=\frac{coefficient\: of\: x}{coefficient \: of\: x^{3}} =\frac{c}{a}\]

\[Product \: of\: zeroes\: (\alpha \beta\gamma )=\frac{constantt\: term}{coefficient \: of\: x^{3}} =\frac{-d}{a}\]

Division Algorithm:-

            Dividend = Divisor x Quotient + Remainder  or

                    p(x) = g(x) x q(x) + r(x)

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Polynomials basic concepts-cbse mathematics