I apologize in advance for asking a homework question, but I have genuinely no idea on how to approach part b). The question is as follows:
(a) Show that the polynomial expression $x^4 -x^2 + x +\frac{5}{4}$ can be written in the form $(x^2-a)^2 + (x-b)^2$
(b) Hence, determine the number of real roots of $y=6x^5 - 10x^3 +15x^2 + 300x +30$
I know how to do part a) and got $(x^2-1)^2+(x+\frac{1}{2})^2=x^4 -x^2 + x +\frac{5}{4}$, but I am unsure of how to do part b) at all. It would be great if in your answer you could explain some of the logic and reasoning behind each step.
Thank you
An interesting observation to make is that when the polynomial is differentiated, a very nice term comes out $$ \begin{align} y^\prime &= 30x^4 -30x^2 +30x +300 \\ &=30\left(x^4 -x^2 +x +10\right) \\ &=30\left((x^2-1)^2 +\left(x+\frac{1}{2}\right)^2 +\frac{35}{4}\right)>0\tag{from part (a)} \end{align} $$ Therefore the function $y=6x^5 - 10x^3 +15x^2 + 300x +30$ is monotonically increasing (strictly increasing) for all $x\in R$. This suggests that the curve is of the sort:
A curve of this form can only have one root