06 - Let's Recap - Polynomials - Class 10 - Maths Class 10 Notes | EduRev

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Class 10 : 06 - Let's Recap - Polynomials - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


Polynomials 
 
Polynomial: An algebraic expression of the form p(x) = a
0
 + a
1
x + a
2
x
2
 + --- + a
n
x
n
 
where a
n
?0 is called a polynomial in variable x of degree n. Where a
0 
, a
1
, ----- a
n
 
are real numbers and each power of x is a non-negative integer. 
Example:- 2x
2
 – 5x + 1 is a polynomial of degree 2. 
Note: vx + 3 is not a polynomial. 
 
Degree of a Polynomial: The highest power of x in a polynomial p(x) is called the 
degree of polynomial. 
 
Different types of Polynomial: 
 
 Constant Polynomial: A polynomial of degree zero is called a constant 
polynomial and it is of the form of p(x) = k  
 
 Linear Polynomial: A polynomial of degree 1 is called a linear polynomial 
and it is of the form p(x) = ax + b, where a, b are real numbers and a
n
?0. 
For example: 3x – 3 , 5x, etc. 
 
 Quadratic Polynomial: A polynomial of degree 2 is called a Quadratic 
Polynomial and it is of the form of p(x) = ax
2
 + bx + c , where a, b, c are real 
numbers and a
n
?0. 
For example: 2x
2
 +x - 1  
 
 Cubic Polynomial: A polynomial of degree 3 is called a Cubic Polynomial 
and it is of the form of p(x) = ax
3
 + bx
2
 + cx + d , where a, b, c, d are real 
numbers and a
n
?0.  
For example: x
3 
– 1 
 
 Bi-quadratic Polynomial: A polynomial of degree 4 is called a Bi-
quadratic Polynomial and it is of the form of p(x) = ax
4
 + bx
3
 + cx
2
 + dx + e, 
where a, b, c, d and e are real numbers and a
n
?0. 
 
Value of a Polynomial: The value of a polynomial p(x) at x =  in the given 
polynomial and is denoted by p(x). 
 
 
 
Page 2


Polynomials 
 
Polynomial: An algebraic expression of the form p(x) = a
0
 + a
1
x + a
2
x
2
 + --- + a
n
x
n
 
where a
n
?0 is called a polynomial in variable x of degree n. Where a
0 
, a
1
, ----- a
n
 
are real numbers and each power of x is a non-negative integer. 
Example:- 2x
2
 – 5x + 1 is a polynomial of degree 2. 
Note: vx + 3 is not a polynomial. 
 
Degree of a Polynomial: The highest power of x in a polynomial p(x) is called the 
degree of polynomial. 
 
Different types of Polynomial: 
 
 Constant Polynomial: A polynomial of degree zero is called a constant 
polynomial and it is of the form of p(x) = k  
 
 Linear Polynomial: A polynomial of degree 1 is called a linear polynomial 
and it is of the form p(x) = ax + b, where a, b are real numbers and a
n
?0. 
For example: 3x – 3 , 5x, etc. 
 
 Quadratic Polynomial: A polynomial of degree 2 is called a Quadratic 
Polynomial and it is of the form of p(x) = ax
2
 + bx + c , where a, b, c are real 
numbers and a
n
?0. 
For example: 2x
2
 +x - 1  
 
 Cubic Polynomial: A polynomial of degree 3 is called a Cubic Polynomial 
and it is of the form of p(x) = ax
3
 + bx
2
 + cx + d , where a, b, c, d are real 
numbers and a
n
?0.  
For example: x
3 
– 1 
 
 Bi-quadratic Polynomial: A polynomial of degree 4 is called a Bi-
quadratic Polynomial and it is of the form of p(x) = ax
4
 + bx
3
 + cx
2
 + dx + e, 
where a, b, c, d and e are real numbers and a
n
?0. 
 
Value of a Polynomial: The value of a polynomial p(x) at x =  in the given 
polynomial and is denoted by p(x). 
 
 
 
 
 
 
 
Graph of Polynomial:  
 
 Graph of a linear polynomial p(x) = ax + b is a straight line 
 Graph of a quadratic polynomial p(x) = ax
2
 + bx + c is a parabola which 
open upwards like ? if  a > 0 
 Graph of a quadratic polynomial p(x) = ax
2
 + bx + c is a parabola which 
open downwards like n if  a < 0 
 In general, a polynomial p(x) of degree n crosses the x- axis, at most n 
points. 
 
Zeroes of a polynomial:  
 
 ? is said to be zero of a polynomial p(x) if p(?)= 0. The zeros of polynomial  
p(x) are actually the X- coordinates of the points where the graph of y = p(x) 
intersects the X-axis. 
 A polynomial of degree “n” can have at most n zeros. For example: A linear 
polynomial has only one zero, a quadratic polynomial has two zeroes and a 
 Cubic polynomial has three zeroes. 
 
Division Algorithm for Polynomials: 
 
 If p(x) and g(x) are any two polynomials with g(x) ?0 then we can find 
polynomials q(x) and r(x) such that 
         P(x) = g(x) x q(x) + r (x) 
i.e. Dividend = Divisor x Quotient + Remainder 
where, r(x) = 0 or degree of r (x) < degree of g(x). This result is known as 
the division algorithm for polynomials. 
 
 
 
 
 
 
 
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