As Sudip Sinha has pointed out
Because all of the coefficients are real numbers, any imaginary zeros must occur in conjugate pairs.
and then divide
But it's quicker to consider the possible rational zero for
There is an error in your question. The root should be
The expression has all real coefficients, so by the Complex Conjugate Roots Theorem (https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem), we have that the other complex root is
Clearly, the third root (say
We will try to get this factor in the expression.
We may write:
As an intro, I think that the root should be
On that basis my answer is :
By using the idea of complex conjugates and some other cool tricks.
One interesting fact about complex roots is that they never occur alone.They always occur in conjugate pairs.
And since there is just one more root left, we can call that root
It is not a complex number because, complex roots always occur in pairs.
And since this is the last of the
In the end the factors of
NB : Note that the difference between a root and a factor is that :
- A root could be
But the corresponding factor would be
The second trick is that, by factoring
Next, expand the braces,
Next, we equate this to the original polynomial
Since the two polynomials are identical, we equate the coefficients of
Actually,we just need to pick one equation and to solve it for
Equating the constant terms,
Hence the last root is