Chapter 2. Polynomials
Exercise 2.2
Q1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x2 – 2x – 8
Solution:
Using Middle term splitting method:
x2 – 4x + 2x – 8 = 0
⇒ x(x – 4) + 2(x – 4) = 0
⇒ x(x – 4) + 2(x – 4) = 0
⇒ (x – 4) (x + 2) = 0
⇒ x – 4 = 0 x + 2 = 0
⇒ x = 4, x = – 2
Zeroes; α = 4 β = – 2
Verifying the relationship between the zeroes and the coefficients.
a = 1, b = – 2, and c = – 8
(ii) 4s2 – 4s + 1
Solution:
4s2– 4s + 1 = 0
⇒ 4s2 – 2s – 2s + 1 = 0
⇒ 2s (2s – 1) –1(2s – 1) = 0
⇒ (2s – 1) (2s – 1) = 0
⇒ 2s – 1= 0, 2s – 1= 0
⇒ 2s = 1, 2s = 1
Coefficients
a = 4, b = – 4, c = 1
Relationship between zeroes and coefficients
L.H.S = R.H.S
L.H.S = R.H.S
Hence verified the relationship between coefficients and the zeroes in both cases.
(iii) 6x2 – 3 – 7x
Solution:
⇒ 6x2 – 3 – 7x = 0
⇒ 6x2 – 7x – 3 = 0 (After rearranging the equation)
⇒ 6x2 – 9x + 2x – 3 = 0
⇒ 3x (2x – 3) +1 (2x – 3) = 0
⇒ (2x – 3) (3x + 1) = 0
⇒ 2x – 3 = 0, 3x + 1 = 0
⇒ 2x = 3, 3x = - 1
Hence verified the relationship between coefficients and the zeroes in both cases.
(iv) 4u2 + 8u
Solution: 4u(u + 2) = 0
4u = 0, u + 2 = 0
u = 0, u = – 2
Zeroes: a = 0, b = – 2
Coefficients:
a = 4, b = 8, c = 0
Verifying relationship between zeroes and coefficients
⇒ 0 = 0
⇒ L.H.S = R.H.S
Hence verified, the relationship between coefficients and zeroes in both cases.
(v) t2 – 15
Solution:
t2 – 15 = 0
t2 = 15
t = ±√15
t = √15, t = – √15
Zeroes: a = √15, b = – √15
Coefficients a = 1, b = 0, c = – 15
Verifying relationship between zeroes and Coefficients
Hence verified, the relationship between coefficients and zeroes in both cases.
(vi) 3x2 – x – 4
Solution:
3x2 – x – 4 = 0
⇒ 3x2 + 3x – 4x – 4 = 0
⇒ 3x (x + 1) – 4 (x + 1) = 0
⇒ (x + 1) (3x – 4) = 0
⇒ x + 1 = 0, 3x – 4 = 0
⇒ x = –1 and 3x = 4
L.H.S= R.H.S
Hence verified, the relationship between coefficients and zeroes in both cases.
Q2. . Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
Hence, required quadratic polynomial is 4x2 – x – 4
Hence, required quadratic polynomial is 3x2 –3 x + 1
Hence, required quadratic polynomial is x2 + √5
Hence, required quadratic polynomial is x2 – x + 1
Hence, required quadratic polynomial is 4X2 + x + 1
Hence, required quadratic polynomial is x2 – 4 + 1