Final answer:
Factoring the polynomial x^4 - 4x^3 + 6x^2 - 4x - 255 with a given root of 1 + 4i involves using its conjugate 1 - 4i, then applying synthetic or long division to reduce it to a quadratic equation. The quadratic can be solved or factored further to find other roots. Knowing calculator operations for roots is essential for these types of problems.
Step-by-step explanation:
To factor the polynomial x^4 - 4x^3 + 6x^2 - 4x - 255 given one root is 1 + 4i, we must use the fact that complex roots of polynomials with real coefficients come in conjugate pairs. Hence, 1 - 4i is also a root of the polynomial. To factor this polynomial, we can apply synthetic division or long division using the divisor (x - (1 + 4i))(x - (1 - 4i)), which simplifies to x^2 - 2x + 17.
After dividing the polynomial by x^2 - 2x + 17, we will get a quadratic polynomial. If the quadratic is factorable, we can factor it further, otherwise, we can solve for its roots using the quadratic formula.
Moreover, for other equilibrium problems involving square roots, cube roots, or higher, knowing how to perform these operations on a calculator is essential. This can be crucial when solving quadratic equations or simplifying expressions to solve for an unknown value. If unsure about these calculations, ask your instructor for assistance.