Book: RD Sharma - Mathematics
Chapter: 2. Polynomials
Subject: Maths - Class 10th
Q. No. 1 of Exercise 2.1
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Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi)
(vii) 
(viii) 
(i) 
⇒ x2 - 4x + 2x – 8
⇒ x (x - 4) + 2 (x - 4)
For zeros of f(x), f(x) = 0
(x + 2) (x - 4) = 0
x = -2, 4
Therefore zeros of the polynomial are -2 & 4
Sum of zeros = -2 + 4 = 2 = -(-2) = 2 = 
Product of zeros = -2 × 4 = -8 = -8 = 
(ii) 
⇒ 4s2 -2s - 2s + 1
⇒ 2s (2s - 1) -1 (2s - 1)
For zeros of g(s), g(s) = 0
(2s - 1) (2s - 1) = 0
s = 
Therefore zeros of the polynomial are 
, 
Sum of zeros = 
+ 
= 1 = -(-
)= 1 = 
Product of zeros = 
× 
= 
= 
= 
(iii) 
For zeros of h(t), h(t) = 0
t2 = 15
t = ± √15
Therefore zeros of t = √15 & -√15
Sum of zeros = √15 + (- √15) = 0 = 0 = 
Product of zeros = 
× 
= -15 = -15 = 
(iv) f(x) = 
x
⇒ 6x2 - 7x -3
⇒ 6x2 - 9x + 2x - 3
⇒ 3x(2x - 3) +1(2x - 3)
⇒ (3x + 1) (2x - 3)
For zeros of f(x), f(x) = 0
⇒ (3x + 1) (2x - 3) = 0
x = 
Therefore zeros of the polynomial are 
Sum of zeros = 
= 
=
= 
= 
Product of zeros = 
× 
= 
= 
= 
(v) P (x) = x2 + 3√2x - √2x - 6
For zeros of p(x), p(x) = 0
⇒ x (x + 3√2) -√2 (x + 3√2) = 0
⇒ (x - √2) (x + 3√2) = 0
x = √2, -3√2
Therefore zeros of the polynomial are √2 & -3√2
Sum of zeros = √2 -3√2 = -2√2 = -2√2 = 
Product of zeros = √2 × -3√2 = -6 = -6 = 
(vi) q (x) = √3x2 + 10x + 7√3
⇒ √3x2 + 10x + 7√3
⇒ √3x2 + 7x + 3x + 7√3
⇒ √3x (x +
) + 3 (x +
)
⇒ (√3x + 3) (x +
)
For zeros of Q(x), Q(x) = 0
(√3x + 3) (x +
) = 0
X = 
, 
Therefore zeros of the polynomial are 
, 
Sum of zeros = 
+ 
= 
Product of zeros = = 
× 
= 
7 = 
(vii) f(x) = x2 - (√3 + 1)x + √3
f(x) = x2 - √3x - x + √3
f(x) = x(x - √3) -1(x - √3)
f(x) = (x - 1) (x - √3)
For zeros of f(x), f(x) = 0
(x - 1) (x - √3) = 0
X = 1, √3
Therefore zeros of the polynomial are 1 & √3
Sum of zeros = 
Product of zeros = 1 × √3 = √3 = √3 = 
(viii) g(x) = a(x2 + 1) – x(a2 + 1)
g(x) = ax2 - a2x – x + a
g(x) = ax(x - a) -1(x - a)
g(x) = (ax - 1) (x - a)
For zeros of g(x), g(x) = 0
(ax - 1) (x - a) = 0
X = 
, a
Therefore zeros of the polynomial are 
& a
Sum of zeros = 
Product of zeros = 
× a = 1 = 1 = 
, find the value of 
, find the value of 
.
, find the value of
, find the value of 
, find the value of 
, find the value of 
, find the value of
, find the value of 
is negative of the other, find the value of k.
is equal to their product, find the value of k.
is equal to 144, find the value of p.
, prove that 
, show that 
and
, find a quadratic polynomial having α and
, find a quadratic polynomial whose zeros are 
, find a quadratic polynomial whose zeros are 
, from a polynomial whose zeros are 
, find a polynomial whose roots are (i) 
(ii) 
, then evaluate:
(ii) 
(vi) 
