Use the rational zeros theorem to find all the real zeros of

Use the rational zeros theorem to find all the real zeros of the polynomial function Use the zeros to factor f over the real numbers f(x)=4x^4 + 9x^3 + 14x^2 +27x + 6 Find the real zeros of f Select the correct choice below and fill in any answer boxes in your choice (Simplify your answer Type an exact answer using fractions as needed Use integers or fractions for any rational numbers in the expression. Use a comma to separate answers as needed There are no real zeros Use the real zeros to factor f. (Simplify your answer Type you answer in factored form Type an exact answer using radicals as needed Use integers or fractions for any rational numbers in the expression)

Solution

Solution

             Use   the    Rational   Theorem   of zeroes to    find    the   real zeroes of the polynomial

             function

             Given   function   is  

                 f(x) = 4x^4 + 9x^3 + 14x^2 + 27x + 6

            Since ,

                f(-2) = 0

           Therefore , (x+2)    is     a factor    of    f(x)

            We   Can   find   rest    of    the factors   as

               [ 4x^4 + 9x^3 + 14x^2 + 27x + 6 ] / [ x + 2]

             = (4x+1)( x^2 + 3)

            Therefore , All    the   factors    are

            = (4x + 1)( x^2 + 3)( x + 2)

   So , All zeroes are

x = - 1/4 , x = -2    and    x = (sqrt3)i

   Now, All    real zeroes are

x = -1/4 ,    x = -2

   and    All the real zeroes to factor    F is

            =    [ 4x^4 + 9x^3 + 14x^2 + 27x + 6 ]

=    (4x + 1)(x^2 + 3)( x + 2)