Video Transcript
Given that two plus 𝑖 root three
is a root of 𝑥 to the power of four minus 12𝑥 cubed plus 55𝑥 squared minus 120𝑥
plus 112 equals zero, find all the roots.
We have a complex root of a
polynomial with real coefficients. And so, by the conjugate root
theorem, its complex conjugate two minus 𝑖 root three is also a root of this
equation. We found two roots then. But how do we find the other
two? The factor theorem gives us two
linear factors of our quartic from these two roots. What can we say about the remaining
factor? Well, it must be quadratic so that
distributing our right-hand side gives a quartic as on the left. And the roots of this unknown
quadratic factor are the remaining roots of our quartic equation. Let’s set about finding this
quadratic factor then.
We multiply the two linear factors
together to get a known quadratic factor. There are a couple of identities
which reduce the amount of calculation required. Now, we can just divide our quartic
by this known quadratic factor to find the unknown quadratic factor. Alternatively, we can distribute on
the right-hand side multiplying every term in the first set of parentheses by every
term in the second set and then collecting like terms. We can then compare
coefficients.
For example, the coefficient of 𝑥
to the four tells us that 𝑎 is one. We make the substitution and then
compare the coefficients of 𝑥 cubed, finding that 𝑏 minus four is negative 12. And so, 𝑏 is negative eight. Again, making the substitution, we
then compare the coefficients of 𝑥 squared, finding that 𝑐 is 16. And making the substitution really
does make the right-hand side equal to the left. We found the unknown quadratic
factor then. It’s one 𝑥 squared minus eight 𝑥
plus 16. And we can factor this as 𝑥 minus
four squared. It has repeated root of four. And so, the original quartic must
do too.
Having factored our quartic, we
read off the roots. The roots are four, repeated root
with multiplicity two; two plus 𝑖 root three, the complex root that we were given;
and its complex conjugate two minus 𝑖 root three.
