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