Question

Show that (x + 1) is a factor of the polynomial `x^3 - x^2 - (2 + sqrt(2))x + sqrt(2)`.

Answer 1

Let p(x) = `x^3 - x^2 - (2 + sqrt(2))x + sqrt(2)`

By factor theorem, (x + 1) will be the factor of p(x) if p(– 1) = 0.

Now, p(– 1) = `(-1)^3 - (-1)^2 - (2 + sqrt(2)) (-1) + sqrt(2)`

= `-1 - 1 + 2 + sqrt(2) + sqrt(2)` 

= `2sqrt(2)`

∵ p(– 1) ≠ 0

Hence, (x + 1) is not a factor of p(x).