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# Class 10 NCERT Solutions – Chapter 2 Polynomials – Exercise 2.2

• Last Updated : 25 Feb, 2021

### Question 1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x2 â€“ 2x â€“ 8

x2 â€“ 2x â€“ 8 = x2 â€“ 4x + 2x â€“ 8

= x (x â€“ 4) + 2(x â€“ 4)

= (x – 4) (x + 2)

Therefore, zeroes of equation x2 â€“ 2x â€“ 8 are (4, -2)

Sum of zeroes is equal to [4 â€“ 2]= 2 = -(-2)/1

i.e. = -(Coefficient of x) / (Coefficient of x2)

Product of zeroes is equal to 4 Ã— (-2) = -8 =-(8)/1

i.e.= (Constant term) / (Coefficient of x2)

(ii) 4s2 â€“ 4s + 1

4s2 â€“ 4s + 1 = 4s2 â€“ 2s â€“ 2s +1

= 2s(2s â€“ 1) â€“ 1(2s – 1)

= (2s â€“ 1) (2s â€“ 1)

Therefore, zeroes of  equation 4s2 â€“ 4s +1 are (1/2, 1/2)

Sum of zeroes is equal to [(1/2) + (1/2)] = 1 = -4/4

i.e.= -(Coefficient of s) / (Coefficient of s2)

Product of zeros is equal to [(1/2) Ã— (1/2)] = 1/4

i.e.= (Constant term) / (Coefficient of s2 )

(iii) 6x2 â€“ 3 â€“ 7x

6x2 â€“ 3 â€“ 7x = 6x2 â€“ 7x â€“ 3

= 6x2 â€“ 9x + 2x â€“ 3

= 3x(2x â€“ 3) + 1(2x â€“ 3)

= (3x + 1) (2x – 3)

Therefore, zeroes of equation 6x2 â€“ 3 â€“ 7x are (-1/3, 3/2)

Sum of zeroes is equal to -(1/3) + (3/2) = (7/6)

i.e.= -(Coefficient of x) / (Coefficient of x2)

Product of zeroes is equal to -(1/3) Ã— (3/2) = -(3/6)

i.e.= (Constant term) / (Coefficient of x2 )

(iv) 4u2 + 8u

4u2 + 8u = 4u(u + 2)

Therefore, zeroes of equation 4u2 + 8u are (0, -2).

Sum of zeroes is equal to [0 + (-2)] = -2 = -(8/4)

i.e. = -(Coefficient of u) / (Coefficient of u2)

Product of zeroes is equal to 0 Ã— -2 = 0 = 0/4

i.e. = (Constant term) / (Coefficient of u2 )

(v) t2 â€“ 15

t2 â€“ 15

â‡’ t2 = 15 or t = Â±âˆš15

Therefore, zeroes of equation t2 â€“ 15 are (âˆš15, -âˆš15)

Sum of zeroes is equal to [âˆš15 + (-âˆš15)] = 0 = -(0/1)

i.e.= -(Coefficient of t) / (Coefficient of t2)

Product of zeroes is equal to âˆš15 Ã— (-âˆš15) = -15 = -15/1

i.e. = (Constant term) / (Coefficient of t2 )

(vi) 3x2 â€“ x â€“ 4

3x2 â€“ x â€“ 4 = 3x2 â€“ 4x + 3x â€“ 4

= x(3x – 4) + 1(3x – 4)

= (3x â€“ 4) (x + 1)

Therefore, zeroes of equation 3x2 â€“ x â€“ 4 are (4/3, -1)

Sum of zeroes is equal to (4/3) + (-1) = (1/3) = -(-1/3)

i.e. = -(Coefficient of x) / (Coefficient of x2)

Product of zeroes is equal to (4/3) Ã— (-1) = (-4/3)

i.e. = (Constant term) / (Coefficient of x2)

### Question 2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i) 1/4, -1

Let two zeroes be Î±, Î²

âˆ´ Sum of zeroes = Î± + Î²

âˆ´ Product of zeroes = Î±Î²

Given, Sum of zeroes = Î± + Î² = 1/4

Product of zeroes = Î± Î² = -1

âˆ´ If Î± and Î² are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x2 â€“ (Î± + Î²)x + Î±Î² = 0

x2 â€“ (1/4)x +(-1) = 0

4x2 â€“ x – 4 = 0

âˆ´ 4x2 â€“ x â€“ 4 is the quadratic polynomial.

(ii) âˆš2, 1/3

Let two zeroes be Î±, Î²

âˆ´ Sum of zeroes = Î± + Î²

âˆ´ Product of zeroes = Î±Î²

Given Sum of zeroes = Î± + Î² =âˆš2

Product of zeroes = Î±Î² = 1/3

âˆ´ If Î± and Î² are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:

x2 â€“ (Î± + Î²)x + Î±Î² = 0

x2 â€“ (âˆš2)x + (1/3) = 0

3x2 – 3âˆš2x + 1 = 0

âˆ´ 3x2 – 3âˆš2x + 1 is the quadratic polynomial.

(iii) 0, âˆš5

Let two zeroes be Î±, Î²

âˆ´ Sum of zeroes = Î± + Î²

âˆ´ Product of zeroes = Î±Î²

Given, Sum of zeroes = Î± + Î² = 0

Product of zeroes = Î±Î² = âˆš5

âˆ´ If Î± and Î² are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x2 â€“ (Î± + Î²)x + Î±Î² = 0

x2 â€“ (0)x + âˆš5 = 0

âˆ´ x2 + âˆš5 is the quadratic polynomial.

(iv) 1, 1

Let two zeroes be Î±, Î²

âˆ´ Sum of zeroes = Î± + Î²

âˆ´ Product of zeroes = Î±Î²

Given, Sum of zeroes = Î± + Î² = 1

Product of zeroes = Î±Î² = 1

âˆ´ If Î± and Î² are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x2 â€“ (Î± + Î²)x + Î±Î² = 0

x2 â€“ x + 1 = 0

âˆ´ x2 â€“ x + 1 is the quadratic polynomial.

(v) -1/4, 1/4

Let two zeroes be Î±, Î²

âˆ´ Sum of zeroes = Î± + Î²

âˆ´ Product of zeroes = Î±Î²

Given, Sum of zeroes = Î± + Î² = -1/4

Product of zeroes = Î± Î² = 1/4

âˆ´ If Î± and Î² are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x2 â€“ (Î± + Î²)x + Î±Î² = 0

x2 â€“ (-1/4)x + (1/4) = 0

4x2 + x + 1 = 0

âˆ´ 4x2 + x + 1 is the quadratic polynomial.

(vi) 4, 1

Let two zeroes be Î±, Î²

âˆ´ Sum of zeroes = Î± + Î²

âˆ´ Product of zeroes = Î±Î²

Given, Sum of zeroes = Î± + Î² = 4

Product of zeroes = Î±Î² = 1

âˆ´ If Î± and Î² are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:

x2 â€“ (Î± + Î²)x + Î±Î² = 0

x2 â€“ 4x + 1 = 0

âˆ´ x2 â€“ 4x + 1 is the quadratic polynomial.

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