Here is an exploration of the polynomial and its derivative using Maple.

>	with(plots):

Warning, the name changecoords has been redefined

>	f:=7*x^4-24*x^3+10*x^2+24*x-17:

>	fGraph:=plot(f,x=-2..3,y=-30..30,thickness=2):

>	display(fGraph);

>	factor(f);

>	solve({f=0},{x});

>	fprime:=diff(f,x);

>	factor(fprime);

>	solve({fprime=0},{x});

>	eval(f,x=1);

>	eval(f,x=2);

>	eval(f,x=-3/7);

>

evalf(%);

>	fprimeGraph:=plot(fprime,x=-2..3,y=-30..30,thickness=2,color=blue):

>	display(fGraph,fprimeGraph);

In the polynomial (f2) below the y-intercept has been changed (from what it was in f) which leads to a need to approximate the x-intercepts.  Notice that f2 cannot be factored over the

set of rational numbers.

>	f2:=7*x^4-24*x^3+10*x^2+24*x-22:

>	f2graph:=plot(f2,x=-2..3,y=-30..30,thickness=2):

>	display(f2graph);

>	factor(f2);

>	fsolve({f2=0},{x});

>	f2prime:=diff(f,x);

>	factor(f2prime);

>	solve({f2prime=0},{x});

>	eval(f2,x=1);

>	eval(f2,x=2);

>	eval(f2,x=-3/7);

>

evalf(%);

>	f2primeGraph:=plot(f2prime,x=-2..3,y=-30..30,thickness=2,color=blue):

>	display(f2graph,f2primeGraph);

In the polynomial f3 the coefficient of x has also been changed so that the zeroes of the derivative also had to be approximated.  Notice that both f3 and f3prime cannot be factored

over the set of rational numbers.

>	f3:=7*x^4-24*x^3+12*x^2+24*x-22:

>	f3graph:=plot(f3,x=-2..3,y=-30..30,thickness=2):

>	display(f3graph);

>	factor(f3);

>	fsolve({f3=0},{x});

>	f3prime:=diff(f3,x);

>	factor(f3prime);

>	fsolve({f3prime=0},{x});

>	eval(f3,x=-.4113053730);

>	eval(f3,x=1.116909911);

>	eval(f3,x=1.865824034);

>	f3primeGraph:=plot(f3prime,x=-2..3,y=-30..30,thickness=2,color=blue):

>	display(f3graph,f3primeGraph);

>