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# Determine which of the following polynomials has (x plus 1) a factor : x ^3 - x ^2 - (2 + root 2)x + root 2

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

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

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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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