Question 1
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x² − 5x + 6
Step 1: Given polynomial: x² − 5x + 6
Step 2: Factorise:
x² − 5x + 6 = (x − 2)(x − 3)
Zeroes: 2 and 3
Verification:
Sum = 2 + 3 = 5 = −(−5)/1 ✔
Product = 2 × 3 = 6 = 6/1 ✔
(ii) 9x² − 12x + 4
Factorise:
9x² − 12x + 4 = (3x − 2)²
Zeroes: 2/3, 2/3
Sum = 4/3 = −(−12)/9 ✔
Product = 4/9 ✔
(iii) 6x² − 5x − 6
Factorise:
6x² − 5x − 6 = (3x + 2)(2x − 3)
Zeroes = −2/3 , 3/2
Sum = (−2/3 + 3/2) = 5/6 ✔
Product = −1 ✔
(iv) 4x² + 12x
Factorise:
4x(x + 3)
Zeroes = 0, −3
Sum = −3 ✔
Product = 0 ✔
(v) t² − 20
Factorise:
(t − √20)(t + √20)
Zeroes = √20 , −√20
Sum = 0 ✔
Product = −20 ✔
(vi) 3x² − 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: x² − (sum)x + product
Polynomial = x² − (1/2)x − 2
Multiplying by 2:
2x² − x − 4
(ii) Sum = √3 , Product = 1/4
Polynomial = x² − √3 x + 1/4
(iii) Sum = 0 , Product = 6
Polynomial = x² + 6
(iv) Sum = 2 , Product = 1
Polynomial = x² − 2x + 1
(v) Sum = −1/2 , Product = 1/3
Polynomial = x² + (1/2)x + 1/3
(vi) Sum = 5 , Product = 6
Polynomial = x² − 5x + 6