Question

Find the quadratic polynomial, for the zeroes α, β given in each case. 

i) 2, -1 

ii) √3, -√3

iii) 1/4 , -1

iv) 1/2 , 3/2

Answer 1

i) Let the polynomial be ax² + bx + c, a ≠ 0 and its zeroes be α and β.

Here α = 2 and β = – 1 

Sum of the zeroes = α + β = 2 + (-1) = 1 

Product of the zeroes = αβ = 2 × (-1) = -2 

Therefore the quadratic polynomial ax² + bx + c is x² – (α + β)x + αβ = [x² – x – 2] 

the quadratic polynomial will be x² – x – 2.

ii) Let the zeroes be α = √3 and β = -√3 

Sum of the zeroes = α + β 

= √3 + (-√3) = 0 

Product of the zeroes = αβ = √3 × (-√3) = -3 

∴ The quadratic polynomial 

ax² + bx + c is [x² – (α + β)x + αβ] 

= [x² – 0.x + (-3)] = [x² – 3] 

the quadratic polynomial will be x² – 3.

iii) Let the zeroes be α = 1/4 and β = -1 

Sum of the zeroes = α + β

=1/4 + (-1) = 1+(-4)/4 = -3/4

 Product of the zeroes = αβ

= 1/4 x (-1) = -1/4

∴ The quadratic polynomial

ax² + bx + c is [x² – (α + β)x + αβ]

= \([x^2-(\frac{-3}{4})+(\frac{-1}{4})]\)

the quadratic polynomial will be 4x² + 3x – 1.

iv) Let the zeroes be α = 1/2 and β = 3/2

Sum of the zeroes = α + β 

= 1/2 + 3/2 = 1+3/2 = 4/2 = 2 

Product of the zeroes = αβ 

= 1/2 × 3/2 = 3/4

∴ The quadratic polynomial 

ax² + bx + c is [x² – (α + β)x + αβ] 

= [x² – 2x + (3/4)] 

the quadratic polynomial will be 4x² – 8x + 3.