Let p ( x ) = 4 x 3 − 4 x 2 + 5 x + 4, how can I show that the quadratic factor of p(x) is positive for all real value of x ?

Question

Let   p  (  x  )  =  4      x    3    −  4      x    2    +  5  x  +  4, how can I show that the quadratic factor of p(x) is positive for all real value of x ?

Izabella Henson · Accepted Answer

If you have found (2x+1) as a factor, then we have  4      x    3    −  4      x    2    +  5  x  +  4  =  (  2  x  +  1  )  (  a      x    2    +  b  x  +  c  )Expanding   (  2  x  +  1  )  (  a      x    2    +  b  x  +  c  ), we get  2  a      x    3    +  (  a  +  2  b  )      x    2    +  (  b  +  2  c  )  x  +  c  =  4      x    3    −  4      x    2    +  5  x  +  4Comparing coefficients, we get that                    2        a                            =        4                            a        +        2        b                            =        −        4                            b        +        2        c                            =        5                            c                            =        4            Hence, we get the quadratic factor as  (  2      x    2    −  3  x  +  4  )Now note that the quadratic factor can be written as follows.                    2                  x          2                −        3        x        +        4                            =        2                  (                      x            2                    −                                    3              2                                x          +          2          )                =        2                  (                      x            2                    −          2          ⋅          x          ⋅                                    3              4                                +                                    (                                                3                  4                                            )                        2                    −                                    (                                                3                  4                                            )                        2                    +          2          )                                                  =        2                  (                                                                                          (                    x                    −                                                                  3                        4                                                              )                                    2                                ⏟                                                    Square is non-negative                                +                                    23              16                                )                ≥                              23            8                              and hence is strictly non-negative.

Let p(x) = 4x^3 - 4x^2 + 5x + 4, how can I show that the quadratic factor of p(x) is positive for all real value of x ?

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