1.(iv) Determine which of the following polynomials has (x + 1) a factor :

    (iv)    x^3 - x^2 -(2 + \sqrt2)x + \sqrt2

Answers (1)

Zero of polynomial   (x + 1)  is -1.

If  (x + 1)  is a factor of polynomial    p(x)=x^3 - x^2 -(2 + \sqrt2)x + \sqrt2  

Then, p(-1)  must be equal to zero 

Now,

\Rightarrow p(-1)=(-1)^3-(-1)^2-(2+\sqrt2)(-1)+\sqrt2

\Rightarrow p(-1)=-1-1+2+\sqrt2+\sqrt2 = 2\sqrt2\neq 0

Therefore,  (x + 1)  is not a factor of polynomial p(x)=x^3 - x^2 -(2 + \sqrt2)x + \sqrt2

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