Solutions:

Concept: When a polynomial is zeroed, the values of x that meet the equation y = f are considered zeros (x). There is a f(x) function, and the zeros of a polynomial are all of the values of x for which the y value equals zero. The degree of the equation y = f determines the number of zeros in a polynomial and vice versa (x).

Calculation:

• x²–2x –8

Factorising the polynomial using splitting method:

⇒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)

• 4s²–4s+1

Factorising the polynomial using splitting method:

⇒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 )

• 6x²–3–7x

Factorising the polynomial using splitting method:

⇒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 )

• 4u²+8u

Factorising the polynomial using splitting method:

⇒ 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 )

• t²–15

Factorising the polynomial using splitting method:

⇒ 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 )

• 3x²–x–4

Factorising the polynomial using splitting method:

⇒ 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 )