X^4-2x^3+6x^2-18x-27 given 3i is a solution, find all solutions

  x ∈ {-1, 3, 3i, -3i}

Step-by-step explanation:

Polynomials with real coefficients have complex roots in conjugate pairs. That means -3i is also a solution to this quartic. Then two factors are ...

  (x -3i)(x +3i) = (x^2 +9)

Dividing the given polynomial by this using your favorite method, you find the other quadratic factor to be ...

  (x^2 -2x -3) = (x +1)(x -3)

The solutions are the values of x that make these factors zero, so are x = -1, and x = 3.

The full set of solutions to this quartic is ...

  x ∈ {-1, 3, -3i, 3i}

Comment on dividing by x^2+9

Since there is no x term in this divisor, the first two terms of the quotient will be (x^4 -2x^3)/x^2 = x^2 -2x. The last term of the quotient will be -27/9 = -3. Thus, we can do the long division with hardly any effort to find ...

  x^4 -2x^3 +6x^2 -18x -27 = (x^2 -2x -3)(x^2 +9)

  = (x +1)(x -3)(x -3i)(x +3i)

The above makes the assumption that there is no remainder, a reasonable assumption for a problem of this nature. In any event, checking the factorization by multiplying it out shows it to be correct.

add 5 subtract 1

step-by-step explanation: