Question

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i)1/4 , -1 (ii) √2 , 1/3​

Answer 1

Answer:

i) α + β = -1/4, αβ = 1/4

thus p(x) = x² - (α + β)x + αβ = 0

• x² - (-1/4)x + 1/4 = 0
• x² + 1/4x + 1/4 = 0
• 1/4(4x² + x + 1) = 0

ii) α + β = √2, αβ = 1/3

thus p(x) = x² - (α + β)x + αβ = 0

• x² - (√2)x + 1/3 = 0
• x² - √2x + 1/3 = 0
• 1/3(3x² - 3√2x + 1) = 0

Answer 2

[i] 1/4(4x² + x + 1)

[ii] 1/3(3x² - 3√2x + 1)

Justification:

Let α, β be the zeros of polynomial.

[i] We have, α + β = - 1/4 & αβ = 1/4

Thus, polynomial is

p(x) = x² - (α + β)x + αβ

• x² - (-1/4)x + 1/4
• x² + 1/4x + 1/4
• 1/4(4x² + x + 1)

[ii] We have, α + β = √2 & αβ = 1/3

Thus, polynomial is

p(x) = x² - (α + β)x + αβ

• x² - √2x + 1/3
• 1/3(3x² - 3√2x + 1)