A quadratic polynomial has one of its zeros 1 + √5 and it satisfies

1.	A quadratic polynomial has one of its zeros 1 + √5 and it satisfies p(1) = 2. Find the quadratic polynomial.
Answer» Given α = 1 + √5 So, β = 1 – √5 α + β = 2; αβ = 1² - (-√5)² = 1 - 5 = -4 The quadratic polynomial is p(x) = x² – (α + β)x + αβ p(x) = k(x² – 2x – 4) p(1) = k(1 – 2 – 4) = -5k Given p (1) = 2 -5k = 2 k = -2/5 ∴ p(x) = (-2/5)(x² - 2x - 4)

A quadratic polynomial has one of its zeros 1 + √5 and it satisfies p(1) = 2. Find the quadratic polynomial.

Answer»

Given α = 1 + √5

So, β = 1 – √5

α + β = 2; αβ = 1² - (-√5)² = 1 - 5 = -4

The quadratic polynomial is

p(x) = x² – (α + β)x + αβ

p(x) = k(x² – 2x – 4)

p(1) = k(1 – 2 – 4) = -5k

Given p (1) = 2

-5k = 2

k = -2/5

∴ p(x) = (-2/5)(x² - 2x - 4)