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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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FAQs on RS Aggarwal Solutions: Polynomials (Exercise 2A) - Mathematics (Maths) Class 10
| 1. What are polynomials and how are they used in mathematics? |  |
Ans. Polynomials are mathematical expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and exponentiation. They are used in various mathematical fields such as algebra, calculus, and number theory to solve equations, graph functions, and model real-world phenomena.
| 2. How do you determine the degree of a polynomial? | |
Ans. The degree of a polynomial is the highest power of the variable in the expression. To determine the degree, identify the term with the highest power of the variable and that will be the degree of the polynomial. For example, in the polynomial 3x^2 + 5x + 1, the highest power of x is 2, so the degree of the polynomial is 2.
| 3. What is the factor theorem and how is it used to find the factors of a polynomial? | |
Ans. The factor theorem states that if a polynomial f(x) has a factor (x - a), where 'a' is a constant, then f(a) = 0. This means that if plugging in the value of 'a' into the polynomial results in zero, then (x - a) is a factor of the polynomial. By using the factor theorem, we can find the factors of a polynomial by testing different values of 'a' and checking if f(a) = 0.
| 4. Can a polynomial have more than one factor? | |
Ans. Yes, a polynomial can have multiple factors. Factors are the expressions that divide evenly into the polynomial, resulting in zero when plugged into the polynomial. For example, if a polynomial f(x) has factors (x - a) and (x - b), where 'a' and 'b' are constants, then both (x - a) and (x - b) are factors of the polynomial.
| 5. How do you add and subtract polynomials? | |
Ans. To add or subtract polynomials, you combine like terms. Like terms are terms that have the same variable and exponent. For example, to add the polynomials 3x^2 + 5x + 1 and 2x^2 - 4x + 3, you add the coefficients of the like terms: (3x^2 + 2x^2) + (5x - 4x) + (1 + 3) = 5x^2 + x + 4. Similarly, for subtraction, you subtract the coefficients of the like terms.
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