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Question

Given that x5 is a factor of the cubic polynomial x335x2+13x35 find all the zeroes of the polynomial.

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Solution

x minus square root of 5 is a factor of x cubed minus 3 square root of 5 x squared plus 13 x minus 3 square root of 5

So, let's divide:

x minus square root of 5 ) x cubed minus 3 square root of 5 x squared plus 13 x minus 3 square root of 5 ( x squared minus 2 square root of 5 x plus 3
x cubed minus square root of 5 x squared
-___+_____
negative 2 square root of 5 x squared space plus 13 x
negative 2 square root of 5 x squared plus 10 x
+_______-_____
3 x minus 3 square root of 5
3 x minus 3 square root of 5
-___+_____
0 0

So,
x cubed minus 3 square root of 5 x squared plus 13 x minus 3 square root of 5 = (x minus square root of 5)(x squared minus 2 square root of 5 x plus 3)

Now, x squared minus 2 square root of 5 x plus 3
x squared plus left parenthesis square root of 2 minus square root of 5 right parenthesis x space minus left parenthesis square root of 2 plus square root of 5 right parenthesis x plus 3 x space left parenthesis space x space plus space left parenthesis square root of 2 minus square root of 5 right parenthesis right parenthesis space minus left parenthesis square root of 2 plus square root of 5 right parenthesis left parenthesis x plus left parenthesis square root of 2 minus square root of 5 right parenthesis right parenthesis left square bracket x plus left parenthesis square root of 2 minus square root of 5 right parenthesis right square bracket space left square bracket space left parenthesis space x space minus space left parenthesis square root of 2 plus square root of 5 right parenthesis right square bracket
x= negative left parenthesis square root of 2 minus square root of 5 right parenthesis , left parenthesis square root of 2 plus square root of 5 right parenthesis end root


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