CBSE Class 10 Mathematics Chapter 2 - Polynomials Exercise - 2.2

 CBSE Class 10 Mathematics  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
Ans.	x2 – 2x – 8 ⇒ x2 – 4x + 2x – 8             (Factor out the common term) ⇒ x (x – 4) + 2 (x – 4) ⇒ (x – 4) (x + 2) So, zeroes of polynomial is x – 4 = 0 or x = 4   = α x + 2 = 0 or x = –2   =  β We know that the relationship between the zeroes and the coefficients is: The sum of Zeros = α + β  =   The product of zeros = α β  =   In given polynomial x2 – 2x – 8;  a = 1, b = –2 and c = –8 Put these value: 4 + (–2) =   = 2                   It is verified 4 × (–2) =   = – 8                    It is verified
(ii)	4s2 – 4s + 1
Ans.	4s2 – 4s + 1 ⇒ 4s2 – 2s – 2s + 1                (Factor out the common term) ⇒ 2s (2s – 1) – 1 (2s – 1) ⇒ (2s – 1) (2s –1) So, zeroes of polynomial is 2s – 1 = 0 or 2s = 1  = s = ½ = α 2s – 1 = 0 or 2s = 1  = s = ½ =  β We know that the relationship between the zeroes and the coefficients is: The sum of Zeros = α + β  =   The product of zeros = α β  =   In given polynomial 4s2 – 4s + 1;  a = 4, b = –4 and c = 1 Put these value: ½ + ½ =   = 1                   It is verified ½ × ½ =   =  ¼                    It is verified
(iii)	6x2  – 3 – 7x
Ans.	6x2  – 3 – 7x We can write this equation as 6x2 – 7x – 3 ⇒ 6x2 – 9x + 2x – 3                  (Factor out the common term) ⇒ 3x (2x – 3) +1 (2x – 3) ⇒ (3x+1) (2x – 3) So, zeroes of polynomial is 3x + 1 = 0 ⇒  3x = –1 ⇒  x =  = α 2x – 3 = 0 ⇒  2x = 3   ⇒ x =   = β We know that the relationship between the zeroes and the coefficients is: The sum of Zeros = α + β  =   The product of zeros = α β  =   In given polynomial 6x2 – 7x – 3;  a = 6, b = –7 and c = –3 Put these value:  +  ⇒   =                     It is verified  ×  =   =                     It is verified
(iv)	4u2 + 8u
Ans.	4u2 + 8u 4u (u+2)          (Factor out the common term) So, zeroes of polynomial is 4u = 0       = α u + 2 = –2 = β We know that the relationship between the zeroes and the coefficients is: The sum of Zeros = α + β  =   The product of zeros = α β  =   In given polynomial 4u2 + 8u;  a = 4, b = 8 and c = 0 Put these value: 0 + (–2) =  = –2          It is verified 0 × (–2) =  = 0                It is verified
(v)	t2 – 15
Ans.	t2 – 15 ⇒ t2 = 15 ⇒ t = ± √15          (by taking the square root of both sides) So, zeroes of polynomial is √15  and  – √15 We know that the relationship between the zeroes and the coefficients is: The sum of Zeros = α + β  =   The product of zeros = α β  =   In given polynomial t2 – 15;  a = 1, b = 0 and c = –√15 Put these value: (√15) + (–√15) =  = 0          It is verified √15 × (–√15) =  = –√15     It is verified
(vi)	3x2 – x – 4
Ans.	3x2 – x – 4 ⇒ 3x2 – 4x + 3x – 4         (Factor out the common term) ⇒ x (3x – 4) + 1 (3x - 4) ⇒ (3x – 4) (x + 1) So, zeroes of polynomial is 3x – 4 = 0 ⇒  3x = 4 ⇒  x =  = α x + 1 = 0 ⇒  x = –1  =  β We know that the relationship between the zeroes and the coefficients is: The sum of Zeros = α + β  =   The product of zeros = α β  =   In given polynomial 3x2 – x – 4;  a = 3, b = –1 and c = –4 Put these value:   + (–1) ⇒  ⇒   =            It is verified  × (–1) ⇒     =         It is verified
Q2.	Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i)	¼  ,  –1
Ans.	We know that, The sum of Zeros = α + β = ¼ The product of zeros = α β  = –1 In a quadratic polynomial, if α and β are zeroes, then it can be written as: x2 – (α + β) x + αβ = 0 Put value from the question: x2 – ¼ x + (–1) = 0 Multiply both side by 4 we get: 4x2 – x – 4 = 0           This is quadratic polynomial
(ii)	√2 , ⅓
Ans.	We know that, The sum of Zeros = α + β = √2 The product of zeros = α β  =  ⅓ In a quadratic polynomial, if α and β are zeroes, then it can be written as: x2 – (α + β) x + αβ = 0 Put value from the question: x2 – √2 x + ⅓ = 0 Multiply both side by 3 we get: 3x2 – 3√2 x + 1 = 0          This is quadratic polynomial
(iii)	0, √5
Ans.	We know that, The sum of Zeros = α + β = 0 The product of zeros = α β  =  √5 In a quadratic polynomial, if α and β are zeroes, then it can be written as: x2 – (α + β) x + αβ = 0 Put value from the question: x2 – (0) x + √5 = 0 x2 + √5 = 0          This is quadratic polynomial
(iv)	1 , 1
Ans.	We know that, The sum of Zeros = α + β = 1 The product of zeros = α β  =  1 In a quadratic polynomial, if α and β are zeroes, then it can be written as: x2 – (α + β) x + αβ = 0 Put value from the question: x2 – 1 x + 1= 0 x2 – x +1 = 0          This is quadratic polynomial
(v)	–¼  , ¼
Ans.	We know that, The sum of Zeros = α + β = –¼ The product of zeros = α β  =  ¼ In a quadratic polynomial, if α and β are zeroes, then it can be written as: x2 – (α + β) x + αβ = 0 Put value from the question: x2 – (–¼) x + ¼ = 0 x2 + ¼ x + ¼ = 0 Multiply both side by 4 we get: 4(x2 + ¼ x + ¼) = 0×4 4x2 + x + 1 = 0       This is quadratic polynomial
(vi)	4  , 1
Ans.	We know that, The sum of Zeros = α + β = 4 The product of zeros = α β  = 1 In a quadratic polynomial, if α and β are zeroes, then it can be written as: x2 – (α + β) x + αβ = 0 Put value from the question: x2 – 4 x + 1 = 0       This is quadratic polynomial