# Question #27939

##### 3 Answers

As Sudip Sinha has pointed out

#### Explanation:

Because all of the coefficients are real numbers, any imaginary zeros must occur in conjugate pairs.

Therefore,

If

and then divide

But it's quicker to consider the possible rational zero for

#### Explanation:

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

Note

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** :

#### Explanation:

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*.

So if

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