Question

given that x-root 5 is a factor of the cubic polynomial x^3-3root5x^2-3root5, find all the zeroes of the polynomial
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Answer 1

given that x-√5 is a factor of the cubic polynomial x3-3√5x2+13x-3√5

            x-√5 ) x3-3√5x2+13x-3√5 ( x2 -2√5x + 3     
                      x3- √5x2                         ( substract )
                     -------------------------------
                       - 2√5x2+13x
                       - 2√5x2+10x                  ( substract )
                      ------------------------------
                                   3x - 3√5
                                   3x - 3√5             ( substract )
                                  ------------------------
                                        0
∴ The quotient is x2 -2√5x + 3 = 0

Using roots of quadratic formula

a = 1,  b = 2√5,  c = 3

x = (-b ± √(b2 - 4ac) ) / 2a

x = (2√5 ± √((2√5)2 - 12) ) / 2
 
∴ the other zeros are x =  √5 + √2 and √5 - √2

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