Question

solve:x^4-6x^3+11x^2-10x+2=0,given that 2-√3 is aroot
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Answer 1

Given :  x⁴ - 6x³ + 11x²  - 10x  + 2  , 2-√3 is a root

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To find : remaining roots

Solution:

2-√3  is a root

Hence one root would be 2 + √3

(x - ( 2-√3) )(x - (2+√3))

= ( x - 2 + √3)(x - 2 - √3)

= ( x - 2)² - (√3)²

= x² - 4x  + 4 - 3

= x² - 4x  + 1

                       x² -2x +2

x² - 4x  + 1 _|   x⁴ - 6x³ + 11x²  - 10x  + 2  |_

                       x⁴ - 4x³  + x²

                    ____________

                            -2x³ + 10x²  - 10x  + 2

                             -2x³ +8x²  - 2x

                        _________________

                                      2x²  - 8x  + 2

                                     2x²  - 8x  + 2

                                _______________

                                            0

   x² -2x +2  is other factor

x =(2 ± √-4)/2  = 1  ± i

Roots are

2± √3   ,  1  ± i

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