Given that 𝑖 is one of the roots
of the equation 𝑥 cubed minus five 𝑥 squared plus 𝑥 minus five equals zero, find
the other two roots.
The factor theorem tells us that 𝑥
minus 𝑖 is one of the factors of this polynomial. And so, we could divide the
left-hand side by this factor to get the quadratic 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐,
which we could then solve. But we can make things easier for
ourselves by using the conjugate root theorem which tells us that the complex
conjugate of 𝑖 must also be a root as we’re dealing with a polynomial with real
coefficients. And so, 𝑥 plus 𝑖 again by the
factor theorem must be a factor of this polynomial.
Multiplying the two known factors
together, we get 𝑥 squared plus one. And we can distribute again. And comparing coefficients, we can
see that 𝑚 is one and 𝑛 is negative five. We can substitute these values
then, factoring our cubic as 𝑥 minus one times 𝑥 plus one times 𝑥 minus five. Remember that we’re looking for the
roots of this equation. And we can just read them off from
the factored form. We find that they are five,
negative 𝑖, and 𝑖. For this problem, the conjugate
root theorem saved us some working, but wasn’t essential.