Find all real zeros of the polynomial. use quadratic formula. P ( x ) = x 5 − 8 x 4 + 7 x 3 + 30 x 2 + 2 x − 12

Question

Find all real zeros of the polynomial. use quadratic formula.   P  (  x  )  =      x    5    −  8      x    4    +  7      x    3    +  30      x    2    +  2  x  −  12

Franklin Frey · Accepted Answer

Consider the polynomial   P  (  x  )  =      x    5    −  8      x    4    +  7      x    3    +  30      x    2    +  2  x  −  12Factor the polynomial as  P  (  x  )  =      x    3    (      x    2    −  8      x    2    +  7  )  +  30      x    2    +  2  x  −  12    =  (  x  +  1      )    2    (  x  −  6  )  (      x    2    −  4  x  +  2  )Now to find real zeros, set P(x)=0 and solve for x.  P  (  x  )  =  0    (  x  +  1      )    2    (  x  −  6  )  (      x    2    −  4  x  +  2  )  =  0    (  x  +  1      )    2    =  0    x  =  −  1Now solve the equation       x    2    −  4  x  +  2  =  0 using quadratic formula   x  =            −      b      ±                        b          2                −        4        a        c                    2      a       as  x  =            (      −      4      )      ±              (        −        4                  )          2                −        4        (        1        )        (        2        )                    2      (      1      )          =            4      ±              8              2      =            4      ±      2              2              2      =  2  ±      2  Therefore, the real zeros are   [  −  1  ,  6  ,  2  −      2    ,  2  +      2    ]

Find all real zeros of the polynomial. use quadratic formula. P(x) = x^5 − 8x^4 + 7x^3 + 30x^2 + 2x − 12

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