Question 1
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x� - 5x + 6
Step 1: Given polynomial: x� - 5x + 6
Step 2: Factorise:
x� - 5x + 6 = (x - 2)(x - 3)
Zeroes: 2 and 3
Verification:
Sum = 2 + 3 = 5 = -(-5)/1 ?
Product = 2 � 3 = 6 = 6/1 ?
(ii) 9x� - 12x + 4
Factorise:
9x� - 12x + 4 = (3x - 2)�
Zeroes: 2/3, 2/3
Sum = 4/3 = -(-12)/9 ?
Product = 4/9 ?
(iii) 6x� - 5x - 6
Factorise:
6x� - 5x - 6 = (3x + 2)(2x - 3)
Zeroes = -2/3 , 3/2
Sum = (-2/3 + 3/2) = 5/6 ?
Product = -1 ?
(iv) 4x� + 12x
Factorise:
4x(x + 3)
Zeroes = 0, -3
Sum = -3 ?
Product = 0 ?
(v) t� - 20
Factorise:
(t - v20)(t + v20)
Zeroes = v20 , -v20
Sum = 0 ?
Product = -20 ?
(vi) 3x� - 5x - 2
Factorise:
(3x + 1)(x - 2)
Zeroes = -1/3 , 2
Sum = 5/3 ?
Product = -2/3 ?
Question 2
Find a quadratic polynomial whose sum and product of zeroes are given.
(i) Sum = 1/2 , Product = -2
General form: x� - (sum)x + product
Polynomial = x� - (1/2)x - 2
Multiplying by 2:
2x� - x - 4
(ii) Sum = v3 , Product = 1/4
Polynomial = x� - v3 x + 1/4
(iii) Sum = 0 , Product = 6
Polynomial = x� + 6
(iv) Sum = 2 , Product = 1
Polynomial = x� - 2x + 1
(v) Sum = -1/2 , Product = 1/3
Polynomial = x� + (1/2)x + 1/3
(vi) Sum = 5 , Product = 6
Polynomial = x� - 5x + 6