Question

Find the quadratic polynomial in each case, with the given numbers as the sum and product of its zeroes respectively. 

i) 1/4 , -1 

ii) √2, 1/3

iii) 0, √5 

iv) 1, 1 

v) –1/4 , 1/4

vi) 4, 1

Answer 1

Let the polynomial be ax² + bx + c and its zeroes be α and β.

i) Here, α + β = 1/4 and αβ = -1 

Thus, the polynomial formed = x² – (sum of the zeroes)x + product of the zeroes 

= x² – (1/4)x – 1 

= x² – x/4 – 1 

The other polynomials are (x² – x/4 – 1) 

then the polynomial is 4x² – x – 4.

ii) Here, α + β = √2 and αβ = 1/3

Thus, the polynomial formed = x² – (sum of the zeroes)x + product of the zeroes 

= x² – (√2)x + 1/3

= x² – √2x + 1/3

The other polynomials are (x² – √2x +1/3) 

then the polynomial is 3x² – 3√2x + 1.

iii) Here, α + β = 0 and αβ = √5 

Thus, the polynomial formed = x² – (sum of the zeroes)x + product of the zeroes 

= x² – (0)x + √5 

= x² + √5

iv) Let the polynomial be ax² + bx + c and its zeroes be α and β. 

Then α + β = 1 = -(-1)/1 = -b/a and 

αβ = 1 = 1/1 = c/a

If a = 4, then b = 1 and c = 1 

∴ One quadratic polynomial which satisfies the given conditions is 4x² + x+ 1.

v) Let the polynomial be ax² + bx + c and its zeroes be α and β. 

Then α + β = -1/4 = -b/a and 

αβ = 1/4 = c/a

If a = 4, then b = 1 and c = 1 

∴ One quadratic polynomial which satisfies the given conditions is 4x² + x + 1.

vi) Let the polynomial be ax² + bx + c and its zeroes be α and β. 

Then α + β = 4 = -(-4)/1= -b/a and 

αβ = 1 = 1/1 = c/a

If a = 1, then b = -4 and c = 1 

∴ One quadratic polynomial which satisfies the given conditions is x² – 4x + 1.