(i) x^{2} – 2x – 8 (ii) 4s^{2} – 4s + 1 (iii) 6x^{2} – 3 – 7x (iv) 4u^{2} + 8u (v) t^{2} – 15 (vi) 3x^{2} – x – 4

**Solution:**

**(i) x ^{2 }– 2x – 8 **

**= **x^{2 }– 4x + 2x – 8

= x(x – 4) + 2(x – 4)

= (x – 4)(x + 2)

Therefore, zeroes of polynomial equation x^{2 }– 2x – 8 are (4, -2)

Sum of zeroes = 4 – 2 = 2 = -(-2)/1 = -(Coefficient of x)/(Coefficient of x^{2})

Product of zeroes = 4 × (-2) = -8 =-(8)/1 = (Constant term)/(Coefficient of x^{2})

**(ii) 4s ^{2 }– 4s + 1 **

= 4s^{2 }– 2s – 2s + 1

= 2s(2s – 1) – 1(2s – 1)

= (2s – 1)(2s – 1)

Therefore, zeroes of polynomial equation 4s^{2 }– 4s + 1 are (1/2, 1/2)

Sum of zeroes = (½) + (1/2) = 1 = -(-4)/4 = -(Coefficient of s)/(Coefficient of s^{2})

Product of zeros = (1/2) × (1/2) = 1/4 = (Constant term)/(Coefficient of s^{2 })

**(iii) 6x ^{2 }– 3 – 7x **

= 6x^{2 }– 7x – 3

= 6x^{2 }– 9x + 2x – 3

= 3x(2x – 3) + 1(2x – 3)

= (3x + 1)(2x – 3)

Therefore, zeroes of polynomial equation 6x^{2 }– 3 – 7x are (-1/3, 3/2)

Sum of zeroes = -(1/3) + (3/2) = (7/6) = -(Coefficient of x)/(Coefficient of x^{2})

Product of zeroes = -(1/3) × (3/2) = -(3/6) = (Constant term) /(Coefficient of x^{2 })

**(iv) 4u ^{2 }+ 8u **

= 4u(u + 2)

Therefore, zeroes of the polynomial equation 4u^{2} + 8u are (0, -2).

Sum of zeroes = 0 + (-2) = -2 = -(8/4) = -(Coefficient of u)/(Coefficient of u^{2})

Product of zeroes = 0 × -2 = 0 = 0/4 = (Constant term)/(Coefficient of u^{2 })

**(v) t ^{2 }– 15 **

= t^{2} = 15 or t = ±√15

Therefore, zeroes of polynomial equation t^{2} –15 are (√15, -√15)

Sum of zeroes = √15 + (-√15) = 0 = -(0/1) = -(Coefficient of t) / (Coefficient of t^{2})

Product of zeroes = √15×(-√15) = -15 = -15/1 = (Constant term) / (Coefficient of t^{2 })

**(vi) 3x ^{2 }– x – 4**

= 3x^{2 }– 4x + 3x – 4

= x(3x – 4) + 1(3x – 4)

= (3x – 4)(x + 1)

Therefore, zeroes of polynomial equation 3x^{2} – x – 4 are (4/3, -1)

Sum of zeroes = (4/3) + (-1) = (1/3) = -(-1/3) = -(Coefficient of x) / (Coefficient of x^{2})

Product of zeroes = (4/3) × (-1) = (-4/3) = (Constant term) /(Coefficient of x^{2 })