Question

Factorise the following:

`x^2 + (a + 1/a)x + 1`

Answer 1

Given expression: `x^2 + (a + 1/a)x + 1`

Step-wise calculation:

1. Rewrite the middle term by splitting the coefficient of x:

`x^2 + ax + x/a + 1`

2. Group the terms as follows:

`x^2 + ax + x/a + 1`

= `x^2 + ax + x/a + a/a`

Note: `a/a = 1`, so adding it keeps the expression unchanged.

3. Now group: `x(x + a) + 1/a(x + a)`

4. Factor out the common factor (x + a):

`(x + a)(x + 1/a)`

The factorization of the expression `x^2 + (a + 1/a) x + 1` is  `(x + a)(x + 1/a)`.