Question

Find the other zeroes of the polynomial 2x²-11x³+13x-7, it being given that two of its zeroes are (3+​√2) and (3-√2).

Answer 1

ol:
If  α and β are zeroes of the polynomial then x2 -(α+ β)x + αβ .

α = (3+√2) and β =  (3 - √2).

α+ β = 6 and  αβ = 7.

∴  x2 - 6x + 7.

Given polynomial 2x4-11x3+ 7x2 + 13x - 7.


                                 x2 - 6x + 7 ) 2x4-11x3+ 7x2 + 13x - 7 ( 2x2 + x -1

                                                    2x4-12x3+14x2  (substract)
                                                  -----------------------------
                                                            x3 - 7x2 + 13x
                                                            x3-  6x2  + 7x   (substract)
                                                           --------------------------------------
                                                                 - x2 + 6x - 7
                                                                 - x2 + 6x - 7  (substract)
                                                                -----------------------------
                                                                             0
∴ The Quotient is 2x2 + x -1

                         = 2x2 + 2x - x  -1

                         = 2x(x + 1)-1( x + 1)
 
                         = ( 2x -1)( x +1).
∴ x = 1/2 , -1 are the other zeros of the polynomial.

2) 
Using remainder theorem f(x) = g(x) (2x-1) + (x+3)

4x3-8x2 + 8x + 1 = g(x) (2x-1) + (x+3)

4x3-8x2 + 7x - 2  = g(x) (2x-1).

g(x) = (4x3-8x2 + 7x - 2) / (2x -1).

              2x -1 ) 4x3-8x2 + 7x - 2 ( 2x2 - 3x + 2
                         4x3-2x2   (substract)
                         ---------------------------------
                              - 6x2 + 7x
                              - 6x2 + 3x   (substract)
                         ---------------------------------
                                        4x - 2
                                        4x - 2  (substract)
                         ---------------------------------
                                           0
∴ g(x) = 2x2 - 3x + 2.
hope it helps............chill
all d best for board exam