Question

If α and β are zeroes of the polynomial p(x) = x² – 2x – 1, then find the value of `1/(2α) + 1/(2β) + 3αβ`.

Answer 1

Given: The polynomial p(x) = x² – 2x – 1 has zeros α and β.

Step-wise calculation:

1. Sum and product of roots for x² – 2x – 1:

α + β = 2, αβ = –1

2. Compute `1/(2α) + 1/(2β) + 3αβ`:

`1/(2α) + 1/(2β)`

= `(1/2)(1/α + 1/β)` 

= `(1/2)((α + β)/(αβ))`

3. Substitute values:

`(1/2)((α + β)/(αβ)) + 3αβ`

= `(1/2)(2/(-1)) + 3(-1)` 

= `(1/2)(-2) - 3`

= –1 – 3

= –4

The value is –4.