Question

Give roots of the following equation 2x^5 -14x^4 +31x^3 -64x^2 +19x +130=0

Answer 1

Answer:

-1,  2,  5,  (1 + i√5)/2,  (1 - i√5)/2

Step-by-step explanation:

Experimenting with putting in a few small values for x in the hope of finding some roots, we find that 2, 5 and -1 are roots.  It follows that (x-2), (x-5) and (x+1) are factors.  Dividing by these gives the factorization:

              2x⁵ - 14x⁴ + 31x³ - 64x² + 19x + 130

          =  (x - 5) (x - 2) (x + 1) (2x² - 2x + 13)

So the remaining roots are those of the quadratic 2x² - 2x + 13, which are

   ( 2 ± √( 2² - 4×2×13 ) ) / 2×2

=  ( 1 ± √( 1 - 26 ) ) / 2

= ( 1 ± i√5 ) / 2.               [ note that these are complex roots ]