Polynomials basic concepts-cbse mathematics
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Types of polynomial On the basis of terms:-
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:-
A polynomial of degree zero is called constant polynomial
For Example:- 3, 5
Polynomial of degree one is called linear polynomial.
For Example:- P(x)= ax + b
Polynomial of degree two is called Quadratic polynomial.
For Example:- P(x)= ax^{2} + bx + c
Polynomial of degree three is called cubic polynomial.
For Example:- P(x) = ax^{3} + bx^{2} + cx + d
Polynomial of degree four is called linear polynomial.
For Example:- P(x) = ax^{4} + bx^{3} + cx^{2} + dx + e
A polynomial with coefficient zero is called zero polynomial
Values of x for which the given polynomial become zero are called the zeroes of the polynomial.
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. |
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
Relationship between zeroes and coefficients of a quadratic polynomial:-
\[Sum \: of\: zeroes\: (\alpha +\beta )=\frac{-\left ( coefficient\:of\: x \right )}{coefficient\: of\: x^{2}}=\frac{-b}{a}\]
Relationship between zeroes and coefficients of a Cubic polynomial:-
\[Sum\; of\; Product\; of\; zeroes\; taken\; two \: at\: a\: time\:(\alpha \beta+\beta \gamma +\gamma \alpha )=\frac{coefficient\: of\: x}{coefficient \: of\: x^{3}} =\frac{c}{a}\]
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Comments
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