Answer:
(x - 2)(x - 5 + 2i)(x - 5 - 2i)
Step-by-step explanation:
If one zero is 2, then one factor is (x - 2).
Use synthetic division here. Taking the coefficients 1, 8, 6 and -52, divide 2 into them:
2 / 1 8 6 -52
2 20 52
-----------------------------
1 10 26 0
Because the remainder is zero, we know that 2 is a root and (x - 2) is a factor, and that the coefficients of the quadratic quotient are 1, 10 and 26.
The discriminant is thus b^2 - 4ac, or 10^2 - 4(1)(26), or 100 - 104, or -4.
Because the discriminant is negative, we know that there will be two factors which themselves are complex conjugates. One such factor is
10 ± √-4 10 ± 2i
x = --------------- = ---------------
2 2
The complex zeros are thus x = 5 ± 2i.
The corresponding factors are (x - 5 + 2i) and (x - 5 - 2i)
and so the polynomial is
(x - 2)(x - 5 + 2i)(x - 5 - 2i)
