RS Aggarwal Solutions: Exercise 2A -Polynomials Notes | EduRev

Mathematics (Maths) Class 10

Class 10 : RS Aggarwal Solutions: Exercise 2A -Polynomials Notes | EduRev

 Page 1


 
Exercise 2A 
1. Find the zeros of the polynomial f(x) = x
2
+ 7x + 12 and verify the relation between its zeroes
and coefficients.
Sol:
x
2
+ 7x + 12 = 0
x
2
+ 4x + 3x + 12 = 0
x(x+4) + 3(x+4) = 0
(x+4) (x+3) = 0
(x + 4) = 0 or (x + 3) = 0
x = 4 or x = 3
Sum of zeroes = 4 + ( 3) = =
Product of zeroes = ( 4) ( 3) = =
2. Find the zeroes of the polynomial f(x) = x
2
and coefficients.
Sol:
x
2
x
2
x = 4 or x = 2
Sum of zeroes = 4 + ( 2) = 2 = =
Product of zeroes = (4) ( 2) = =
3. Find the zeroes of the quadratic polynomial f(x) = x
2
rify the relation
between its zeroes and coefficients.
Sol:
We have:
f(x) = x
2
= x
2
=
= (x 2) (x + 5)
f(x) = 0 (x 2) (x + 5) = 0
x 2 = 0 or x + 5 = 0
x = 2 or x = 5.
So, the zeroes of f(x) are 2 and 5.
Page 2


 
Exercise 2A 
1. Find the zeros of the polynomial f(x) = x
2
+ 7x + 12 and verify the relation between its zeroes
and coefficients.
Sol:
x
2
+ 7x + 12 = 0
x
2
+ 4x + 3x + 12 = 0
x(x+4) + 3(x+4) = 0
(x+4) (x+3) = 0
(x + 4) = 0 or (x + 3) = 0
x = 4 or x = 3
Sum of zeroes = 4 + ( 3) = =
Product of zeroes = ( 4) ( 3) = =
2. Find the zeroes of the polynomial f(x) = x
2
and coefficients.
Sol:
x
2
x
2
x = 4 or x = 2
Sum of zeroes = 4 + ( 2) = 2 = =
Product of zeroes = (4) ( 2) = =
3. Find the zeroes of the quadratic polynomial f(x) = x
2
rify the relation
between its zeroes and coefficients.
Sol:
We have:
f(x) = x
2
= x
2
=
= (x 2) (x + 5)
f(x) = 0 (x 2) (x + 5) = 0
x 2 = 0 or x + 5 = 0
x = 2 or x = 5.
So, the zeroes of f(x) are 2 and 5.
 
Sum of zeroes = 2 + ( 5) = 3 = =
Product of zeroes = 2 × ( 5) = 10 = =
4. Find the zeroes of the quadratic polynomial f(x) = 4x
2
its zeroes and coefficients.
Sol:
We have:
f(x) = 4x
2
= 4x
2
= 4x
2
3
= 2x (2x 3) + 1(2x 3)
= (2x + 1) (2x 3)
f(x) = 0 (2x + 1) (2x 3)= 0
2x + 1= 0 or 2x 3 = 0
x = or x = 
So, the zeroes of f(x) are and  .
Sum of zeroes = + = = = 1 = 
Product of zeroes = × = =
5. Find the zeroes of the quadratic polynomial f(x) = 5x
2
between the zeroes and coefficients of the given polynomial.
Sol:
We have: 
f(x) = 5x
2
= 5x
2
= 5x
2
= 5x
2
= 5x (x 2) + 2(x 2)
= (5x + 2) (x 2)
f(x) = 0 (5x + 2) (x 2) = 0
5x + 2= 0 or x 2 = 0
x = or x = 2
So, the zeroes of f(x) are and 2. 
Sum of zeroes = + 2 = = =
Product of zeroes = × 2 = =
Page 3


 
Exercise 2A 
1. Find the zeros of the polynomial f(x) = x
2
+ 7x + 12 and verify the relation between its zeroes
and coefficients.
Sol:
x
2
+ 7x + 12 = 0
x
2
+ 4x + 3x + 12 = 0
x(x+4) + 3(x+4) = 0
(x+4) (x+3) = 0
(x + 4) = 0 or (x + 3) = 0
x = 4 or x = 3
Sum of zeroes = 4 + ( 3) = =
Product of zeroes = ( 4) ( 3) = =
2. Find the zeroes of the polynomial f(x) = x
2
and coefficients.
Sol:
x
2
x
2
x = 4 or x = 2
Sum of zeroes = 4 + ( 2) = 2 = =
Product of zeroes = (4) ( 2) = =
3. Find the zeroes of the quadratic polynomial f(x) = x
2
rify the relation
between its zeroes and coefficients.
Sol:
We have:
f(x) = x
2
= x
2
=
= (x 2) (x + 5)
f(x) = 0 (x 2) (x + 5) = 0
x 2 = 0 or x + 5 = 0
x = 2 or x = 5.
So, the zeroes of f(x) are 2 and 5.
 
Sum of zeroes = 2 + ( 5) = 3 = =
Product of zeroes = 2 × ( 5) = 10 = =
4. Find the zeroes of the quadratic polynomial f(x) = 4x
2
its zeroes and coefficients.
Sol:
We have:
f(x) = 4x
2
= 4x
2
= 4x
2
3
= 2x (2x 3) + 1(2x 3)
= (2x + 1) (2x 3)
f(x) = 0 (2x + 1) (2x 3)= 0
2x + 1= 0 or 2x 3 = 0
x = or x = 
So, the zeroes of f(x) are and  .
Sum of zeroes = + = = = 1 = 
Product of zeroes = × = =
5. Find the zeroes of the quadratic polynomial f(x) = 5x
2
between the zeroes and coefficients of the given polynomial.
Sol:
We have: 
f(x) = 5x
2
= 5x
2
= 5x
2
= 5x
2
= 5x (x 2) + 2(x 2)
= (5x + 2) (x 2)
f(x) = 0 (5x + 2) (x 2) = 0
5x + 2= 0 or x 2 = 0
x = or x = 2
So, the zeroes of f(x) are and 2. 
Sum of zeroes = + 2 = = =
Product of zeroes = × 2 = =
 
6. Find the zeroes of the polynomial f(x) = 2 + and verify the relation between its
zeroes and coefficients.
Sol:
2 +
2 +
2 ( (
( ) = 0
( ) = 0
x = or x = 
x = × = or x = 
Sum of zeroes = + = =
Product of zeroes = × = =
7. Find the zeroes of the quadratic polynomial 2x
2
the zeroes and the coefficients.
Sol:
f(x) = 2x
2
= 2x
2
= 2x
2
= 2x (x x 3)
= (2x 5) (x 3)
f(x) = 0 (2x 5) (x 3) = 0
2x 5= 0 or x 3 = 0
    x = or x = 3
So, the zeroes of f(x) are and 3. 
Sum of zeroes = + 3 = = =
Product of zeroes = × 3  = =
8. Find the zeroes of the quadratic polynomial 4x
2
between the
zeroes and the coefficients.
Sol:
4x
2
(2x)
2
2(2x)(1) + (1)
2
= 0
Page 4


 
Exercise 2A 
1. Find the zeros of the polynomial f(x) = x
2
+ 7x + 12 and verify the relation between its zeroes
and coefficients.
Sol:
x
2
+ 7x + 12 = 0
x
2
+ 4x + 3x + 12 = 0
x(x+4) + 3(x+4) = 0
(x+4) (x+3) = 0
(x + 4) = 0 or (x + 3) = 0
x = 4 or x = 3
Sum of zeroes = 4 + ( 3) = =
Product of zeroes = ( 4) ( 3) = =
2. Find the zeroes of the polynomial f(x) = x
2
and coefficients.
Sol:
x
2
x
2
x = 4 or x = 2
Sum of zeroes = 4 + ( 2) = 2 = =
Product of zeroes = (4) ( 2) = =
3. Find the zeroes of the quadratic polynomial f(x) = x
2
rify the relation
between its zeroes and coefficients.
Sol:
We have:
f(x) = x
2
= x
2
=
= (x 2) (x + 5)
f(x) = 0 (x 2) (x + 5) = 0
x 2 = 0 or x + 5 = 0
x = 2 or x = 5.
So, the zeroes of f(x) are 2 and 5.
 
Sum of zeroes = 2 + ( 5) = 3 = =
Product of zeroes = 2 × ( 5) = 10 = =
4. Find the zeroes of the quadratic polynomial f(x) = 4x
2
its zeroes and coefficients.
Sol:
We have:
f(x) = 4x
2
= 4x
2
= 4x
2
3
= 2x (2x 3) + 1(2x 3)
= (2x + 1) (2x 3)
f(x) = 0 (2x + 1) (2x 3)= 0
2x + 1= 0 or 2x 3 = 0
x = or x = 
So, the zeroes of f(x) are and  .
Sum of zeroes = + = = = 1 = 
Product of zeroes = × = =
5. Find the zeroes of the quadratic polynomial f(x) = 5x
2
between the zeroes and coefficients of the given polynomial.
Sol:
We have: 
f(x) = 5x
2
= 5x
2
= 5x
2
= 5x
2
= 5x (x 2) + 2(x 2)
= (5x + 2) (x 2)
f(x) = 0 (5x + 2) (x 2) = 0
5x + 2= 0 or x 2 = 0
x = or x = 2
So, the zeroes of f(x) are and 2. 
Sum of zeroes = + 2 = = =
Product of zeroes = × 2 = =
 
6. Find the zeroes of the polynomial f(x) = 2 + and verify the relation between its
zeroes and coefficients.
Sol:
2 +
2 +
2 ( (
( ) = 0
( ) = 0
x = or x = 
x = × = or x = 
Sum of zeroes = + = =
Product of zeroes = × = =
7. Find the zeroes of the quadratic polynomial 2x
2
the zeroes and the coefficients.
Sol:
f(x) = 2x
2
= 2x
2
= 2x
2
= 2x (x x 3)
= (2x 5) (x 3)
f(x) = 0 (2x 5) (x 3) = 0
2x 5= 0 or x 3 = 0
    x = or x = 3
So, the zeroes of f(x) are and 3. 
Sum of zeroes = + 3 = = =
Product of zeroes = × 3  = =
8. Find the zeroes of the quadratic polynomial 4x
2
between the
zeroes and the coefficients.
Sol:
4x
2
(2x)
2
2(2x)(1) + (1)
2
= 0
 
(2x 1)
2
= 0 [ a
2
2ab + b
2
= (a b)
2
]
(2x 1)
2
= 0
x = or x = 
Sum of zeroes = + = 1 = =
Product of zeroes = × = =
9. Find the zeroes of the quadratic polynomial (x
2
) and verify the relation between the zeroes
and the coefficients.
Sol:
We have:
f(x) = x
2
It can be written as x
2
=
= (x + ) (x )
f(x) = 0 (x + ) (x ) = 0
x + = 0 or x = 0
x = or x = 
So, the zeroes of f(x) are and .
Here, the coefficient of x is 0 and the coefficient of is 1.
Sum of zeroes = + = =
Product of zeroes = × = =
10. Find the zeroes of the quadratic polynomial (8x
2
) and verify the relation between the
zeroes and the coefficients.
Sol:
We have:
f(x) = 8x
2
It can be written as 8x
2
= 4 { (1)
2
}
= 4 ( + 1) ( 1)
f(x) = 0 ( + 1) ( 1) = 0
( + 1) = 0 or 1 = 0
x = or x = 
Page 5


 
Exercise 2A 
1. Find the zeros of the polynomial f(x) = x
2
+ 7x + 12 and verify the relation between its zeroes
and coefficients.
Sol:
x
2
+ 7x + 12 = 0
x
2
+ 4x + 3x + 12 = 0
x(x+4) + 3(x+4) = 0
(x+4) (x+3) = 0
(x + 4) = 0 or (x + 3) = 0
x = 4 or x = 3
Sum of zeroes = 4 + ( 3) = =
Product of zeroes = ( 4) ( 3) = =
2. Find the zeroes of the polynomial f(x) = x
2
and coefficients.
Sol:
x
2
x
2
x = 4 or x = 2
Sum of zeroes = 4 + ( 2) = 2 = =
Product of zeroes = (4) ( 2) = =
3. Find the zeroes of the quadratic polynomial f(x) = x
2
rify the relation
between its zeroes and coefficients.
Sol:
We have:
f(x) = x
2
= x
2
=
= (x 2) (x + 5)
f(x) = 0 (x 2) (x + 5) = 0
x 2 = 0 or x + 5 = 0
x = 2 or x = 5.
So, the zeroes of f(x) are 2 and 5.
 
Sum of zeroes = 2 + ( 5) = 3 = =
Product of zeroes = 2 × ( 5) = 10 = =
4. Find the zeroes of the quadratic polynomial f(x) = 4x
2
its zeroes and coefficients.
Sol:
We have:
f(x) = 4x
2
= 4x
2
= 4x
2
3
= 2x (2x 3) + 1(2x 3)
= (2x + 1) (2x 3)
f(x) = 0 (2x + 1) (2x 3)= 0
2x + 1= 0 or 2x 3 = 0
x = or x = 
So, the zeroes of f(x) are and  .
Sum of zeroes = + = = = 1 = 
Product of zeroes = × = =
5. Find the zeroes of the quadratic polynomial f(x) = 5x
2
between the zeroes and coefficients of the given polynomial.
Sol:
We have: 
f(x) = 5x
2
= 5x
2
= 5x
2
= 5x
2
= 5x (x 2) + 2(x 2)
= (5x + 2) (x 2)
f(x) = 0 (5x + 2) (x 2) = 0
5x + 2= 0 or x 2 = 0
x = or x = 2
So, the zeroes of f(x) are and 2. 
Sum of zeroes = + 2 = = =
Product of zeroes = × 2 = =
 
6. Find the zeroes of the polynomial f(x) = 2 + and verify the relation between its
zeroes and coefficients.
Sol:
2 +
2 +
2 ( (
( ) = 0
( ) = 0
x = or x = 
x = × = or x = 
Sum of zeroes = + = =
Product of zeroes = × = =
7. Find the zeroes of the quadratic polynomial 2x
2
the zeroes and the coefficients.
Sol:
f(x) = 2x
2
= 2x
2
= 2x
2
= 2x (x x 3)
= (2x 5) (x 3)
f(x) = 0 (2x 5) (x 3) = 0
2x 5= 0 or x 3 = 0
    x = or x = 3
So, the zeroes of f(x) are and 3. 
Sum of zeroes = + 3 = = =
Product of zeroes = × 3  = =
8. Find the zeroes of the quadratic polynomial 4x
2
between the
zeroes and the coefficients.
Sol:
4x
2
(2x)
2
2(2x)(1) + (1)
2
= 0
 
(2x 1)
2
= 0 [ a
2
2ab + b
2
= (a b)
2
]
(2x 1)
2
= 0
x = or x = 
Sum of zeroes = + = 1 = =
Product of zeroes = × = =
9. Find the zeroes of the quadratic polynomial (x
2
) and verify the relation between the zeroes
and the coefficients.
Sol:
We have:
f(x) = x
2
It can be written as x
2
=
= (x + ) (x )
f(x) = 0 (x + ) (x ) = 0
x + = 0 or x = 0
x = or x = 
So, the zeroes of f(x) are and .
Here, the coefficient of x is 0 and the coefficient of is 1.
Sum of zeroes = + = =
Product of zeroes = × = =
10. Find the zeroes of the quadratic polynomial (8x
2
) and verify the relation between the
zeroes and the coefficients.
Sol:
We have:
f(x) = 8x
2
It can be written as 8x
2
= 4 { (1)
2
}
= 4 ( + 1) ( 1)
f(x) = 0 ( + 1) ( 1) = 0
( + 1) = 0 or 1 = 0
x = or x = 
 
So, the zeroes of f(x) are and 
Here the coefficient of x is 0 and the coefficient of x
2
is
Sum of zeroes = + = = =
Product of zeroes = × = = =
11. Find the zeroes of the quadratic polynomial (5y
2
+ 10y) and verify the relation between the
zeroes and the coefficients.
Sol:
We have,
f (u) = 5u
2
+ 10u
It can be written as 5u (u+2)
f (u) = 0 5u = 0 or u + 2 = 0
u = 0 or u = 2
So, the zeroes of f (u) are 2 and 0.
Sum of the zeroes = 2 + 0 = 2 = = =
Product of zeroes = 2 × 0 = 0 = = =
12. Find the zeroes of the quadratic polynomial (3x
2
) and verify the relation between the
zeroes and the coefficients.
Sol:
3x
2
3x
2
x (3x 4) + 1 (3x 4) = 0
(3x 4) (x + 1) = 0
(3x 4) or (x + 1) = 0
x = or x = 1
Sum of zeroes = + (-1) = =
Product of zeroes = × (-1) = =
13. Find the quadratic polynomial whose zeroes are 2 and -6. Verify the relation between the
coefficients and the zeroes of the polynomial.
Sol:
Let = 2 and = -6
Sum of the zeroes, ( + ) = 2 + (-6) = -4
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