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# 07 - Question bank - Polynomials - Class 10 - Maths Class 10 Notes | EduRev

## Class 10 : 07 - Question bank - Polynomials - Class 10 - Maths Class 10 Notes | EduRev

``` Page 1

POLYNOMIALS

GRAPHS OF POLYNOMIALS

1) Draw a graph of the polynomial f x x
Solution:
Let y = 3 2 x+ now we get different values of y corresponding to the different values of x.

x 0 1 2
y 2 5 8

The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.

RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial
between the zeroes and the coefficients.
Solution:
The given quadratic polynomial can be
Thus the value of
2
12 32 x x + + is zero when
i.e. when 8 x=- or 4 x=- Therefore the zeroes of
Sum of zeroes = -8+(-4)=-12 Product of the zeroes=

POLYNOMIALS
( ) 3 2 f x x = +
now we get different values of y corresponding to the different values of x.

points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial
2
12 32 x x + + and verify the relationship
between the zeroes and the coefficients.
The given quadratic polynomial can be factorised as
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ =
Therefore the zeroes of
2
12 32 x x + + are -8 and -4.
12 Product of the zeroes=-8 x -4=32
now we get different values of y corresponding to the different values of x.
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.

BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
and verify the relationship
12 32 ( 8)( 4) x x x x + + = + +
4.
Page 2

POLYNOMIALS

GRAPHS OF POLYNOMIALS

1) Draw a graph of the polynomial f x x
Solution:
Let y = 3 2 x+ now we get different values of y corresponding to the different values of x.

x 0 1 2
y 2 5 8

The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.

RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial
between the zeroes and the coefficients.
Solution:
The given quadratic polynomial can be
Thus the value of
2
12 32 x x + + is zero when
i.e. when 8 x=- or 4 x=- Therefore the zeroes of
Sum of zeroes = -8+(-4)=-12 Product of the zeroes=

POLYNOMIALS
( ) 3 2 f x x = +
now we get different values of y corresponding to the different values of x.

points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial
2
12 32 x x + + and verify the relationship
between the zeroes and the coefficients.
The given quadratic polynomial can be factorised as
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ =
Therefore the zeroes of
2
12 32 x x + + are -8 and -4.
12 Product of the zeroes=-8 x -4=32
now we get different values of y corresponding to the different values of x.
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.

BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
and verify the relationship
12 32 ( 8)( 4) x x x x + + = + +
4.
POLYNOMIALS

( )
( )
2
Coefficient of x
Coefficient of x
-
=
(12)
1
-
= =-

(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

3) Find the zeroes of the quadratic polynomial
between the zeroes and the coefficients.
Solution:
Using the identity
2 2
( )( ) a b a b a b - = + -
?

v6


Hence, the value of 

v6

is zero when
Hence x = -v6 and x = v6 are the zeroes of
Sum of zeroes 6 6 0 = - =  Product of zeroes
( )
( )
2
Coefficient of x
Coefficient of x
-
0
0
1
= = as the quadratic polynomial x

2
Constant term
Coefficient of x
6
6
1
-
= =-
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

4) Find a quadratic polynomial, the sum and prod
Solution;
Let the quadratic polynomial be ax bx c
Given that
6 a ß + =-
b
a
-
= b a ? =
and       x 5
c
a
a
ß = =    5 c a ? =
If a=1, then b=6 and c=5
Thus, quadratic polynomial which satisfies the given condition is

POLYNOMIALS
(12)
12 = =-
2
Constant term
Coefficient of x

32
32
1
= =

)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find the zeroes of the quadratic polynomial
2
6 x - and verify the relationship
between the zeroes and the coefficients.
( )( ) a b a b a b - = + - we can write,
v6		v6
is zero when 6 0 x+ = or 6 0 x- = i.e. x=-
6 are the zeroes of the quadratic polynomial x
2
- 6
Product of zeroes 6 x - 6 6 = =-
2
– 6 we can write it as x
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5
2
ax bx c + + and its zeroes be a and ß
6 b a ? =
5 c a ? =
quadratic polynomial which satisfies the given condition is
2
6 5 x x + +
2
Constant term
Coefficient of x

6 x=- or 6 x=

write it as x
2
+ 0x - 6
2
Constant term
Coefficient of x

6 and 5 respectively
6 5
Page 3

POLYNOMIALS

GRAPHS OF POLYNOMIALS

1) Draw a graph of the polynomial f x x
Solution:
Let y = 3 2 x+ now we get different values of y corresponding to the different values of x.

x 0 1 2
y 2 5 8

The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.

RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial
between the zeroes and the coefficients.
Solution:
The given quadratic polynomial can be
Thus the value of
2
12 32 x x + + is zero when
i.e. when 8 x=- or 4 x=- Therefore the zeroes of
Sum of zeroes = -8+(-4)=-12 Product of the zeroes=

POLYNOMIALS
( ) 3 2 f x x = +
now we get different values of y corresponding to the different values of x.

points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial
2
12 32 x x + + and verify the relationship
between the zeroes and the coefficients.
The given quadratic polynomial can be factorised as
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ =
Therefore the zeroes of
2
12 32 x x + + are -8 and -4.
12 Product of the zeroes=-8 x -4=32
now we get different values of y corresponding to the different values of x.
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.

BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
and verify the relationship
12 32 ( 8)( 4) x x x x + + = + +
4.
POLYNOMIALS

( )
( )
2
Coefficient of x
Coefficient of x
-
=
(12)
1
-
= =-

(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

3) Find the zeroes of the quadratic polynomial
between the zeroes and the coefficients.
Solution:
Using the identity
2 2
( )( ) a b a b a b - = + -
?

v6


Hence, the value of 

v6

is zero when
Hence x = -v6 and x = v6 are the zeroes of
Sum of zeroes 6 6 0 = - =  Product of zeroes
( )
( )
2
Coefficient of x
Coefficient of x
-
0
0
1
= = as the quadratic polynomial x

2
Constant term
Coefficient of x
6
6
1
-
= =-
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

4) Find a quadratic polynomial, the sum and prod
Solution;
Let the quadratic polynomial be ax bx c
Given that
6 a ß + =-
b
a
-
= b a ? =
and       x 5
c
a
a
ß = =    5 c a ? =
If a=1, then b=6 and c=5
Thus, quadratic polynomial which satisfies the given condition is

POLYNOMIALS
(12)
12 = =-
2
Constant term
Coefficient of x

32
32
1
= =

)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find the zeroes of the quadratic polynomial
2
6 x - and verify the relationship
between the zeroes and the coefficients.
( )( ) a b a b a b - = + - we can write,
v6		v6
is zero when 6 0 x+ = or 6 0 x- = i.e. x=-
6 are the zeroes of the quadratic polynomial x
2
- 6
Product of zeroes 6 x - 6 6 = =-
2
– 6 we can write it as x
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5
2
ax bx c + + and its zeroes be a and ß
6 b a ? =
5 c a ? =
quadratic polynomial which satisfies the given condition is
2
6 5 x x + +
2
Constant term
Coefficient of x

6 x=- or 6 x=

write it as x
2
+ 0x - 6
2
Constant term
Coefficient of x

6 and 5 respectively
6 5
POLYNOMIALS

5) Verify that 2,-3 and 6 are the zeroes of the cubic polynomial
and then verify the relationship between the zeroes and the coefficients.
Solution:
Compare the given polynomial with
a=1 b=-5 c=-12 and d=36.
Now to verify if 2,-3 and 6  are the roots of the polynomial.
i.e. we have to show that 2
?
3 2
(2) 2 5(2) 12(2) 36 8 20 24 36 0 f = - - + = - - + =

3 2
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 f - = - - - - - + =- - + + =

3 2
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 f = - - + = - - + =
Therefore 2,-3 and 6 are the zeroes of the cubic polynomial
Let    =2 a  3 ß =- and 6 ? =
+ 2 ( 3) 6 2 3 6 5 a ß ? + = + - + = - + = =
2( 3) ( 3)6 6(2) 6 18 12 12 a a ß ß? ? + + = - + - + =- - + =- =
2( 3)6 36
d
a
a
ß?
-
= - =- =

6) If a and ß are the zeroes of the polynomial
Solution:
We know that
(
(
Sum of zeroes=
Coefficient of x
-
Product of the zeroes=
Coefficient of x
And hence,
1 1
a ß aß
+ =
?

POLYNOMIALS
3 and 6 are the zeroes of the cubic polynomial
3 2
( ) 5 12 36 f x x x x = - - +
elationship between the zeroes and the coefficients.
Compare the given polynomial with
3 2
ax bx cx d + + + we get,
are the roots of the polynomial.
 0 , 3 = 0 and 6  = 0
(2) 2 5(2) 12(2) 36 8 20 24 36 0 = - - + = - - + =
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 - = - - - - - + =- - + + =
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 = - - + = - - + =
3 and 6 are the zeroes of the cubic polynomial
3 2
5 12 36 x x x - - +
+ 2 ( 3) 6 2 3 6 5
b
a
-
+ = + - + = - + = =
2( 3) ( 3)6 6(2) 6 18 12 12
c
a
+ + = - + - + =- - + =- =

are the zeroes of the polynomial
2
( ) f x ax bx c = + + then find
1 1
a
+
( )
)
2
Coefficient of x
Coefficient of x
-

i.e.
b
a
a
ß
-
+ =
2
Constant term
Coefficient of x
i.e.
c
a
a
ß =
a ß
a ß aß
+
+ =
b
a
c
a
-
=
b
c
-
=
?
1 1 b
c a ß
-
+ =
( ) 5 12 36 f x x x x = - - +
5 12 36
1 1
ß
+
Page 4

POLYNOMIALS

GRAPHS OF POLYNOMIALS

1) Draw a graph of the polynomial f x x
Solution:
Let y = 3 2 x+ now we get different values of y corresponding to the different values of x.

x 0 1 2
y 2 5 8

The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.

RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial
between the zeroes and the coefficients.
Solution:
The given quadratic polynomial can be
Thus the value of
2
12 32 x x + + is zero when
i.e. when 8 x=- or 4 x=- Therefore the zeroes of
Sum of zeroes = -8+(-4)=-12 Product of the zeroes=

POLYNOMIALS
( ) 3 2 f x x = +
now we get different values of y corresponding to the different values of x.

points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial
2
12 32 x x + + and verify the relationship
between the zeroes and the coefficients.
The given quadratic polynomial can be factorised as
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ =
Therefore the zeroes of
2
12 32 x x + + are -8 and -4.
12 Product of the zeroes=-8 x -4=32
now we get different values of y corresponding to the different values of x.
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.

BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
and verify the relationship
12 32 ( 8)( 4) x x x x + + = + +
4.
POLYNOMIALS

( )
( )
2
Coefficient of x
Coefficient of x
-
=
(12)
1
-
= =-

(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

3) Find the zeroes of the quadratic polynomial
between the zeroes and the coefficients.
Solution:
Using the identity
2 2
( )( ) a b a b a b - = + -
?

v6


Hence, the value of 

v6

is zero when
Hence x = -v6 and x = v6 are the zeroes of
Sum of zeroes 6 6 0 = - =  Product of zeroes
( )
( )
2
Coefficient of x
Coefficient of x
-
0
0
1
= = as the quadratic polynomial x

2
Constant term
Coefficient of x
6
6
1
-
= =-
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

4) Find a quadratic polynomial, the sum and prod
Solution;
Let the quadratic polynomial be ax bx c
Given that
6 a ß + =-
b
a
-
= b a ? =
and       x 5
c
a
a
ß = =    5 c a ? =
If a=1, then b=6 and c=5
Thus, quadratic polynomial which satisfies the given condition is

POLYNOMIALS
(12)
12 = =-
2
Constant term
Coefficient of x

32
32
1
= =

)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find the zeroes of the quadratic polynomial
2
6 x - and verify the relationship
between the zeroes and the coefficients.
( )( ) a b a b a b - = + - we can write,
v6		v6
is zero when 6 0 x+ = or 6 0 x- = i.e. x=-
6 are the zeroes of the quadratic polynomial x
2
- 6
Product of zeroes 6 x - 6 6 = =-
2
– 6 we can write it as x
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5
2
ax bx c + + and its zeroes be a and ß
6 b a ? =
5 c a ? =
quadratic polynomial which satisfies the given condition is
2
6 5 x x + +
2
Constant term
Coefficient of x

6 x=- or 6 x=

write it as x
2
+ 0x - 6
2
Constant term
Coefficient of x

6 and 5 respectively
6 5
POLYNOMIALS

5) Verify that 2,-3 and 6 are the zeroes of the cubic polynomial
and then verify the relationship between the zeroes and the coefficients.
Solution:
Compare the given polynomial with
a=1 b=-5 c=-12 and d=36.
Now to verify if 2,-3 and 6  are the roots of the polynomial.
i.e. we have to show that 2
?
3 2
(2) 2 5(2) 12(2) 36 8 20 24 36 0 f = - - + = - - + =

3 2
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 f - = - - - - - + =- - + + =

3 2
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 f = - - + = - - + =
Therefore 2,-3 and 6 are the zeroes of the cubic polynomial
Let    =2 a  3 ß =- and 6 ? =
+ 2 ( 3) 6 2 3 6 5 a ß ? + = + - + = - + = =
2( 3) ( 3)6 6(2) 6 18 12 12 a a ß ß? ? + + = - + - + =- - + =- =
2( 3)6 36
d
a
a
ß?
-
= - =- =

6) If a and ß are the zeroes of the polynomial
Solution:
We know that
(
(
Sum of zeroes=
Coefficient of x
-
Product of the zeroes=
Coefficient of x
And hence,
1 1
a ß aß
+ =
?

POLYNOMIALS
3 and 6 are the zeroes of the cubic polynomial
3 2
( ) 5 12 36 f x x x x = - - +
elationship between the zeroes and the coefficients.
Compare the given polynomial with
3 2
ax bx cx d + + + we get,
are the roots of the polynomial.
 0 , 3 = 0 and 6  = 0
(2) 2 5(2) 12(2) 36 8 20 24 36 0 = - - + = - - + =
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 - = - - - - - + =- - + + =
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 = - - + = - - + =
3 and 6 are the zeroes of the cubic polynomial
3 2
5 12 36 x x x - - +
+ 2 ( 3) 6 2 3 6 5
b
a
-
+ = + - + = - + = =
2( 3) ( 3)6 6(2) 6 18 12 12
c
a
+ + = - + - + =- - + =- =

are the zeroes of the polynomial
2
( ) f x ax bx c = + + then find
1 1
a
+
( )
)
2
Coefficient of x
Coefficient of x
-

i.e.
b
a
a
ß
-
+ =
2
Constant term
Coefficient of x
i.e.
c
a
a
ß =
a ß
a ß aß
+
+ =
b
a
c
a
-
=
b
c
-
=
?
1 1 b
c a ß
-
+ =
( ) 5 12 36 f x x x x = - - +
5 12 36
1 1
ß
+
POLYNOMIALS

7) Find the zeroes of the polynomial
the coefficients
Solution:
Given that,

2
( ) 4 5 4 3 5 f x x x = + -

2
( ) 4 5 10 6 3 5 f x x x x = + - -

( )( )
( ) 2 5 2 5 3 f x x x = + -
The zeroes of the polynomial are given by
i.e.
( )( )
2 5 2 5 3 0 x x + - =
? 2 5 0 x+ = or 2 5 3 0 x- =

5
2
x
-
= or
3
2 5
x=
Hence zeroes are
3
2 5
a = and
a ß + =
3 5 3 5 6 10 4 1
2 2 2 5 2 5 4 5 4 5 5
? ?
+ = - = = =
? ?
? ?
? ?

3 5 3
x
2 4 2 5
aß
? ?
- -
= =
? ?
? ?
? ?

4 1
4 5 5
b
a
- - -
= =
3 5 3
4 5
c
a
- -
= =

b
a
a
ß
-
+ =
c
a
a
ß =

(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

8) If a and ß are the zeroes of the polynomial
Solution:
We know that
(
(
Sum of zeroes=
Coefficient of x
-

POLYNOMIALS
Find the zeroes of the polynomial
2
( ) 4 5 4 3 5 f x x x = + - and verify the relationship between the zeroes and
The zeroes of the polynomial are given by ( ) 0 f x =
2 5 2 5 3 0
2 5 3 0 - =
and
5
2
ß
-
=
3 5 3 5 6 10 4 1
2 2 2 5 2 5 4 5 4 5 5
- - -
+ = - = = =
3 5 3
4
- -
= =
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
are the zeroes of the polynomial
2
( ) f x ax bx c = + +  then find
a ß
ß a
+
( )
)
2
Coefficient of x
Coefficient of x
-
i.e.
b
a
a
ß
-
+ =

POLYNOMIALS
verify the relationship between the zeroes and
2
Constant term
Coefficient of x

a ß
ß a
+
Page 5

POLYNOMIALS

GRAPHS OF POLYNOMIALS

1) Draw a graph of the polynomial f x x
Solution:
Let y = 3 2 x+ now we get different values of y corresponding to the different values of x.

x 0 1 2
y 2 5 8

The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.

RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial
between the zeroes and the coefficients.
Solution:
The given quadratic polynomial can be
Thus the value of
2
12 32 x x + + is zero when
i.e. when 8 x=- or 4 x=- Therefore the zeroes of
Sum of zeroes = -8+(-4)=-12 Product of the zeroes=

POLYNOMIALS
( ) 3 2 f x x = +
now we get different values of y corresponding to the different values of x.

points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.
Since the polynomial is linear we get the graph as a straight line.
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial
2
12 32 x x + + and verify the relationship
between the zeroes and the coefficients.
The given quadratic polynomial can be factorised as
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ =
Therefore the zeroes of
2
12 32 x x + + are -8 and -4.
12 Product of the zeroes=-8 x -4=32
now we get different values of y corresponding to the different values of x.
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.

BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
and verify the relationship
12 32 ( 8)( 4) x x x x + + = + +
4.
POLYNOMIALS

( )
( )
2
Coefficient of x
Coefficient of x
-
=
(12)
1
-
= =-

(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

3) Find the zeroes of the quadratic polynomial
between the zeroes and the coefficients.
Solution:
Using the identity
2 2
( )( ) a b a b a b - = + -
?

v6


Hence, the value of 

v6

is zero when
Hence x = -v6 and x = v6 are the zeroes of
Sum of zeroes 6 6 0 = - =  Product of zeroes
( )
( )
2
Coefficient of x
Coefficient of x
-
0
0
1
= = as the quadratic polynomial x

2
Constant term
Coefficient of x
6
6
1
-
= =-
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

4) Find a quadratic polynomial, the sum and prod
Solution;
Let the quadratic polynomial be ax bx c
Given that
6 a ß + =-
b
a
-
= b a ? =
and       x 5
c
a
a
ß = =    5 c a ? =
If a=1, then b=6 and c=5
Thus, quadratic polynomial which satisfies the given condition is

POLYNOMIALS
(12)
12 = =-
2
Constant term
Coefficient of x

32
32
1
= =

)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find the zeroes of the quadratic polynomial
2
6 x - and verify the relationship
between the zeroes and the coefficients.
( )( ) a b a b a b - = + - we can write,
v6		v6
is zero when 6 0 x+ = or 6 0 x- = i.e. x=-
6 are the zeroes of the quadratic polynomial x
2
- 6
Product of zeroes 6 x - 6 6 = =-
2
– 6 we can write it as x
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5
2
ax bx c + + and its zeroes be a and ß
6 b a ? =
5 c a ? =
quadratic polynomial which satisfies the given condition is
2
6 5 x x + +
2
Constant term
Coefficient of x

6 x=- or 6 x=

write it as x
2
+ 0x - 6
2
Constant term
Coefficient of x

6 and 5 respectively
6 5
POLYNOMIALS

5) Verify that 2,-3 and 6 are the zeroes of the cubic polynomial
and then verify the relationship between the zeroes and the coefficients.
Solution:
Compare the given polynomial with
a=1 b=-5 c=-12 and d=36.
Now to verify if 2,-3 and 6  are the roots of the polynomial.
i.e. we have to show that 2
?
3 2
(2) 2 5(2) 12(2) 36 8 20 24 36 0 f = - - + = - - + =

3 2
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 f - = - - - - - + =- - + + =

3 2
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 f = - - + = - - + =
Therefore 2,-3 and 6 are the zeroes of the cubic polynomial
Let    =2 a  3 ß =- and 6 ? =
+ 2 ( 3) 6 2 3 6 5 a ß ? + = + - + = - + = =
2( 3) ( 3)6 6(2) 6 18 12 12 a a ß ß? ? + + = - + - + =- - + =- =
2( 3)6 36
d
a
a
ß?
-
= - =- =

6) If a and ß are the zeroes of the polynomial
Solution:
We know that
(
(
Sum of zeroes=
Coefficient of x
-
Product of the zeroes=
Coefficient of x
And hence,
1 1
a ß aß
+ =
?

POLYNOMIALS
3 and 6 are the zeroes of the cubic polynomial
3 2
( ) 5 12 36 f x x x x = - - +
elationship between the zeroes and the coefficients.
Compare the given polynomial with
3 2
ax bx cx d + + + we get,
are the roots of the polynomial.
 0 , 3 = 0 and 6  = 0
(2) 2 5(2) 12(2) 36 8 20 24 36 0 = - - + = - - + =
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 - = - - - - - + =- - + + =
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 = - - + = - - + =
3 and 6 are the zeroes of the cubic polynomial
3 2
5 12 36 x x x - - +
+ 2 ( 3) 6 2 3 6 5
b
a
-
+ = + - + = - + = =
2( 3) ( 3)6 6(2) 6 18 12 12
c
a
+ + = - + - + =- - + =- =

are the zeroes of the polynomial
2
( ) f x ax bx c = + + then find
1 1
a
+
( )
)
2
Coefficient of x
Coefficient of x
-

i.e.
b
a
a
ß
-
+ =
2
Constant term
Coefficient of x
i.e.
c
a
a
ß =
a ß
a ß aß
+
+ =
b
a
c
a
-
=
b
c
-
=
?
1 1 b
c a ß
-
+ =
( ) 5 12 36 f x x x x = - - +
5 12 36
1 1
ß
+
POLYNOMIALS

7) Find the zeroes of the polynomial
the coefficients
Solution:
Given that,

2
( ) 4 5 4 3 5 f x x x = + -

2
( ) 4 5 10 6 3 5 f x x x x = + - -

( )( )
( ) 2 5 2 5 3 f x x x = + -
The zeroes of the polynomial are given by
i.e.
( )( )
2 5 2 5 3 0 x x + - =
? 2 5 0 x+ = or 2 5 3 0 x- =

5
2
x
-
= or
3
2 5
x=
Hence zeroes are
3
2 5
a = and
a ß + =
3 5 3 5 6 10 4 1
2 2 2 5 2 5 4 5 4 5 5
? ?
+ = - = = =
? ?
? ?
? ?

3 5 3
x
2 4 2 5
aß
? ?
- -
= =
? ?
? ?
? ?

4 1
4 5 5
b
a
- - -
= =
3 5 3
4 5
c
a
- -
= =

b
a
a
ß
-
+ =
c
a
a
ß =

(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?

8) If a and ß are the zeroes of the polynomial
Solution:
We know that
(
(
Sum of zeroes=
Coefficient of x
-

POLYNOMIALS
Find the zeroes of the polynomial
2
( ) 4 5 4 3 5 f x x x = + - and verify the relationship between the zeroes and
The zeroes of the polynomial are given by ( ) 0 f x =
2 5 2 5 3 0
2 5 3 0 - =
and
5
2
ß
-
=
3 5 3 5 6 10 4 1
2 2 2 5 2 5 4 5 4 5 5
- - -
+ = - = = =
3 5 3
4
- -
= =
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
are the zeroes of the polynomial
2
( ) f x ax bx c = + +  then find
a ß
ß a
+
( )
)
2
Coefficient of x
Coefficient of x
-
i.e.
b
a
a
ß
-
+ =

POLYNOMIALS
verify the relationship between the zeroes and
2
Constant term
Coefficient of x

a ß
ß a
+
POLYNOMIALS

Constant term
Product of the zeroes=
Coefficient of x
( )
2
2 2
2 a ß aß
a ß a ß
ß a aß aß
+ -
+
+ = =
?
2
2 b ac
ac
a ß
ß a
-
+ ==
9) If a and ß are the zeroes of the polynomial
then find the value of k
Solution:
Since a and ß are zeroes of the polynomial.
a +ß =


= -6,   aß =


= k   and a ß
We know that,   ( )
2 2
4 a ß aß a ß + - = -
Substituting the values we get,
(-6)
2
– 4k = 4
2

36 – 4k = 16
36 – 16 = 4k
20 = 4k
Therefore k = 5
10)   Find the zeroes of the polynomial
relationship between the zeroes and coefficients.
Solution:
We have

( )
2 2
2 2
( )
( )
( ) ( ) ( )
( ) ( )( )
f x abx b ac x bc
f x abx b x acx bc
f x bx ax b c ax b
f x ax b bx c
= + - -
= + - -
= + - +
= + -

POLYNOMIALS
2
Constant term
Coefficient of x
i.e.
c
a
a
ß =

2 a ß aß
2
2 2
2 2
2
2
2
b c
b c b ac
a a
a a a
c c c
a a a
- ? ? ? ?
-
-
- ? ? ? ?
? ? ? ?
= = = =
es of the polynomial
2
( ) 6 f x x x k = + + such that a ß - =
are zeroes of the polynomial.
4 a ß - =
( )
2 2
a ß aß a ß + - = -
Find the zeroes of the polynomial
( )
2 2
( ) f x abx b ac x bc = + - - and verify the
elationship between the zeroes and coefficients.
f x abx b ac x bc

2
2 b ac
ac
-
= = = =
4 a ß - =
and verify the
```
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## Mathematics (Maths) Class 10

62 videos|346 docs|103 tests

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