Hi,
could i get some help on this question please
Hey! Firstly, since all the coefficients are real, we know that either ONE of the roots is real, or THREE of the roots are real (as imaginary roots will come in complex conjugates).
The easiest way to answer the first half is by differentiating the function, it proving that it is non-decreasing. So,
For a>0, this derivative is ALWAYS positive. Thus, there will only be one real root, as the graph will never 'turn around'. It's helpful to sketch an example of a non-decreasing function (ie. gradient is always positive), to prove to the marker that you know why this fact results in one real root.
Now, if two roots are equal, we can right them as
Using sum and product rules.
and
We know from above that
so
Now, subbing alpha into the original equation, we get
Not sure where we are going with this; let's keep going?
We are trying to prove that
Subbing it what we've found above;
Which looks like it equals zero! Bit of a round about way of getting there, but hey, it works!