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

$left parenthesis straight x cubed minus straight x squared minus left parenthesis 2 plus square root of 2 right parenthesis straight x plus square root of 2 right parenthesis$

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Solution

left parenthesis straight x cubed minus straight x squared minus left parenthesis 2 plus square root of 2 right parenthesis straight x plus square root of 2 right parenthesis

Let space straight p left parenthesis straight x right parenthesis equals straight x cubed minus straight x squared minus left parenthesis 2 plus square root of 2 right parenthesis left parenthesis negative 1 right parenthesis plus square root of 2 space space space space space equals space minus 1 minus 1 plus 2 plus square root of 2 plus square root of 2 equals 2 square root of 2 not equal to 0

∴ By factor theorem, x + 1 is not a factor of

straight x to the power of 4 minus straight x cubed space left parenthesis 2 plus square root of 2 right parenthesis straight x plus square root of 2

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