P(x)=x^6-2x^5-13x^4+70x^3-160x^2+184x+80

Question

the function P(x)=x^6-2x^5-13x^4+7ultiplicity 2, 1 as a single zero, and -5 as a single zero. find the other zeros in simplest form

Answer 1

There's an error in P(x). It should be P(x)=x⁶-2x⁵-13x⁴+70x³-160x²+184x-80.

The zeroes are 2 (twice), 1, -5. By progressively dividing by these zeroes using synthetic division we can reduce P(x) to a quadratic:

1 | 1 -2 -13  70 -160  184  -80

     1  1  -1  -14    56 -104 | 80

2 | 1 -1 -14  56 -104    80 | 0

     1  2    2 -24    64 | -80

2 | 1  1 -12  32   -40 |    0

     1  2    6 -12 |  40

-5 | 1  3   -6  20 |    0

      1 -5  10 |-20

      1 -2    4 |  0 = x²-2x+4.

x²-2x=-4,

x²-2x+1=-3,

(x-1)²=-3,

x=1±i√3 are the remaining two zeroes: 1+i√3 and 1-i√3.