Question

find all 0 of 2 x 4 - 3 x cube minus 3 X square + 6 x minus 2,i f you know that two of its zeros are √2,-√2

Answer 1

Let p(x) = x4+3x3 - 20x2 - 6x+36

Given √2 and -√2 are the zeroes of the polynomial.

Hence (x + √2) and (x - √2) are the factors of the given polynomial.

(x + √2)(x - √2) = x2 2

On dividing (x4+3x3 - 20x2 - 6x+36) with (x2 2) we the quotient as (x2 +3x 18)

∴ (x4+3x3 - 20x2 - 6x+36) = (x2 2) (x2 +3x 18)

                                                  = (x2 2) (x2 +6x   3x 18)  

                                                  = (x2 2) [x(x +6) 3(x 6)]  

                                                  = (x2 2)(x +6)(x 3)  

∴ (x +6)(x 3)   = 0

⇒ (x +6)= 0 and (x 3) = 0

⇒ x = 6 and x = 3

Therefore the other zeroes are 3 and 6.