<s>[INST] How to determine the remaining roots when two distinct real roots and y-intercept are given A quartic polynomial has 2 distinct real roots at $x=1$ and $x=-3/5$. If the function has a y-intercept at -1 and has $f(2)=2$ and $f(3)=3$, determine the remaining roots and produce an accurate sketch of the function.
 [/INST] Hint: you can sum up everything you know in a system like this:
\begin{cases}
f(1)=0\\
f(-\frac35)=0\\
f(0) = -1\\
f(2) =2\\
f(3) = 3\\
\end{cases}
Then it's very simple to find all the coefficients of $f(x) = ax^4+bx^3+cx^2+dx+e$, because you have a 5-variable system and 5 equations, so the solution is unique! For example, from the 3rd equation we get $f(0) = 0+0+0+0+e=-1 \Rightarrow e=-1$, and so on. After some calculus we obtain: 
\begin{cases}
a=\frac{5}{156}\\
b=-\frac{41}{78}\\
c=\frac{289}{156}\\
d=-\frac{14}{39}\\
e=-1\\
\end{cases}
Finally we manage to find the polynomail, and the associated function: $$f(x) = \frac{5}{156}x^4 -\frac{41}{78}x^3+\frac{289}{156}x^2-\frac{14}{39}x-1$$
So all the real roots are $x_1= -\frac35,x_2= 1,x_3 = 8-2\sqrt{3},x_4= 8+2\sqrt{3}$. Eventually we can sketch a graph of the function: 
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