Points to Remember: Polynomials Notes | EduRev

Mathematics (Maths) Class 10

Class 10 : Points to Remember: Polynomials Notes | EduRev

 Page 1


Polynomials 
Page 2


Polynomials 
POLYNOMIALS 5
a
n
x
n
 + a
n - 1
 + a x
n - 1
 + .......+ a
2
x
2
1
x + a
0
“Polynomial is an algebraic expression with power as whole numbers.”
Polynomial
5
x + 5
2x
2
2x
3
 + 5x
2
 - 3
x
4
 + 3x
2
 - 2x + 1
2x
5
 + 2x
3
 - 3x
2
 - 2x + 6
Degree
0
1
2
3
4
5
Name using 
number of terms
Monomial
Binomial
Monomial
Trinomial
Polynomial of 4 terms 
Polynomial of 5 terms 
No of terms
1
2
1
3
4
5
Linear polynomial has 
only one zero
Relation between zeros & Coecients of polynomial 
(1) Linear polynomial (ax + b)
Zero : a =
-b
a
-b
a
c
a
Real zero exists when graph cuts
x-axis (point of intersection)
(2) Quadratic polynomial (ax
2
 + bx + c)
Sum of zeros ( a + ß) =
Coecient of middle term (x)
Coecent of x
2
Product of zeros ( aß) =
Constant term (3
rd
 term)
Coecient of x
2 {
{
}
}
Quadratic polynomial
has two zeros.
y
x
1 2 3
4 5 6 y
x
y
x
y
x
y
x
y
x
2 cuts on x - axis
2 real roots
3 cuts on x - axis 
3 real roots
1 touch on x - axis
1 root repeated twice
ex : (x + 3)
2
1 touch and 1 cut 
on x - axis.
3 zeros(roots) out of 
which one repeated twice
No cuts on x - axis 
2 non real roots
as it is a graph of 
Quadratic Polynomial
4 cuts on x - axis
4 real roots (zeros)
Algebraic Identity
a
2
 + b
2
 = (a + b)
2
 - 2ab
Quadratic polynomial in root (zeros) form
x
2
 - (
a + ß) x + aß = 0
Polynomials
; where coecients cannot be Zero
Page 3


Polynomials 
POLYNOMIALS 5
a
n
x
n
 + a
n - 1
 + a x
n - 1
 + .......+ a
2
x
2
1
x + a
0
“Polynomial is an algebraic expression with power as whole numbers.”
Polynomial
5
x + 5
2x
2
2x
3
 + 5x
2
 - 3
x
4
 + 3x
2
 - 2x + 1
2x
5
 + 2x
3
 - 3x
2
 - 2x + 6
Degree
0
1
2
3
4
5
Name using 
number of terms
Monomial
Binomial
Monomial
Trinomial
Polynomial of 4 terms 
Polynomial of 5 terms 
No of terms
1
2
1
3
4
5
Linear polynomial has 
only one zero
Relation between zeros & Coecients of polynomial 
(1) Linear polynomial (ax + b)
Zero : a =
-b
a
-b
a
c
a
Real zero exists when graph cuts
x-axis (point of intersection)
(2) Quadratic polynomial (ax
2
 + bx + c)
Sum of zeros ( a + ß) =
Coecient of middle term (x)
Coecent of x
2
Product of zeros ( aß) =
Constant term (3
rd
 term)
Coecient of x
2 {
{
}
}
Quadratic polynomial
has two zeros.
y
x
1 2 3
4 5 6 y
x
y
x
y
x
y
x
y
x
2 cuts on x - axis
2 real roots
3 cuts on x - axis 
3 real roots
1 touch on x - axis
1 root repeated twice
ex : (x + 3)
2
1 touch and 1 cut 
on x - axis.
3 zeros(roots) out of 
which one repeated twice
No cuts on x - axis 
2 non real roots
as it is a graph of 
Quadratic Polynomial
4 cuts on x - axis
4 real roots (zeros)
Algebraic Identity
a
2
 + b
2
 = (a + b)
2
 - 2ab
Quadratic polynomial in root (zeros) form
x
2
 - (
a + ß) x + aß = 0
Polynomials
; where coecients cannot be Zero
Division Algorithm
= × + 
Divide x
4
 + 6x
3
 - 2x
2
 + 7x + 4    by x
2
 + 2x + 1
x
2
 + 2x +1 ) x
4
 + 6x
3
 - 2x
2
 + 7x + 4
x
4
 + 2x
3
 + x
2
 + 0    + 0
4x
3
 - 3x
2
 + 7x  + 4
x
2
x
2
 + 2x + 1  ) 4x
3
 - 3x
2
 + 7x + 4
4x
3
 + 8x
2
 + 4x + 0
-11x
2
 + 3x + 4
4x
x
2
 + 2x + 1  
)
  -11x
2
 + 3x  + 4
-11x
2
 - 22x - 11
25x + 15
-11
Arrange the polynomial in the decreasing
order of their powers (Higher to Lower) 
Step 1
Step 2
Just look at 1
st 
term of divisor and think
of a term that when multiplied with it
gives the 1
st 
terms of dividend
Step 4
Repeat the same procedure until we get
terms with order of degree lower than
that of divisor. In this case ( x
2 
)  
Step 3
Subtract and get the remainder
(take care of signs)
-
- - -
- - - -
+ + +
Dividend Divisor Quotient Remainder
= × +
 x
2 
+ 4x - 11 25x + 15 x
4 
+ 6x
3 
- 2x
2 
+ 7x + 4 x
2 
+ 2x + 1
 Numbers of zeros of a polynomial depend on the 
degree of that polynomial. A Quadratic polynomial can 
have most 2 zeros and cubic can have at most 3 zeros.
?
?
PLEASE KEEP IN MIND
 Zero of a polynomial is a number/ value of
x by which the value of the polynomial becomes zero (0).
Zeros are also known as roots. Don’t get confused in this
 Every polynomial has zeros, sometimes real 
sometimes non-real. And if the polynomial does not cut 
the x-axis and still has zeros then they will be in the form 
of complex numbers (non-real numbers) equal to the 
degree of the polynomial.
?
 Degree of dividend should always be greater
than the divisor (Division algorithm) .
Introduction to polynomials Types and Zeros of Polynomials
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POLYNOMIALS 6
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