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

Solutions: (i) x2–2x –8 ⇒x2– 4x+2x–8 = x(x–4)+2(x–4) = (x-4)(x+2) Therefore, zeroes of polynomial equation x2–2x–8 are (4, -2) Sum of zeroes = 4–2 = 2 = -(-2)/1 = -(Coefficient of x)/(Coefficient of x2) Product of zeroes = 4×(-2) = -8 =-(8)/1 = (Constant term)/(Coefficient of x2) (ii) 4s2–4s+1 ⇒4s2–2s–2s+1 = 2s(2s–1)–1(2s-1) = (2s–1)(2s–1) Therefore, zeroes of polynomial equation 4s2–4s+1 are (1/2, 1/2) Sum of zeroes = (½)+(1/2) = 1 = -4/4 = -(Coefficient of s)/(Coefficient of s2) Product of zeros = (1/2)×(1/2) = 1/4 = (Constant term)/(Coefficient of s2 ) (iii) 6x2–3–7x ⇒6x2–7x–3 = 6x2 – 9x + 2x – 3 = 3x(2x – 3) +1(2x – 3) = (3x+1)(2x-3) Therefore, zeroes of polynomial equation 6x2–3–7x are (-1/3, 3/2) Sum of zeroes = -(1/3)+(3/2) = (7/6) = -(Coefficient of x)/(Coefficient of x2) Product of zeroes = -(1/3)×(3/2) = -(3/6) = (Constant term) /(Coefficient of x2 ) (iv) 4u2+8u ⇒ 4u(u+2) Therefore, zeroes of polynomial equation 4u2 + 8u are (0, -2). Sum of zeroes = 0+(-2) = -2 = -(8/4) = = -(Coefficient of u)/(Coefficient of u2) Product of zeroes = 0×-2 = 0 = 0/4 = (Constant term)/(Coefficient of u2 ) (v) t2–15 ⇒ t2 = 15 or t = ±√15 Therefore, zeroes of polynomial equation t2 –15 are (√15, -√15) Sum of zeroes =√15+(-√15) = 0= -(0/1)= -(Coefficient of t) / (Coefficient of t2) Product of zeroes = √15×(-√15) = -15 = -15/1 = (Constant term) / (Coefficient of t2 ) (vi) 3x2–x–4 ⇒ 3x2–4x+3x–4 = x(3x-4)+1(3x-4) = (3x – 4)(x + 1) Therefore, zeroes of polynomial equation3x2 – x – 4 are (4/3, -1) Sum of zeroes = (4/3)+(-1) = (1/3)= -(-1/3) = -(Coefficient of x) / (Coefficient of x2) Product of zeroes=(4/3)×(-1) = (-4/3) = (Constant term) /(Coefficient of x2 )