Hello, cityismine!
Use synthetic division to show that all of the real zeros of $\displaystyle f(x) \:=\: x^33x^24x+5$
lie in the interval $\displaystyle [2,4]$ I'm not sure why Synthetic Divison is used here . . .
We find that: .$\displaystyle f(2) \:=\:812+8+5 \:=\:7$
. . The graph is below the xaxis.
We find that: .$\displaystyle f(0) \:=\: 0  0  9 + 5 \:=\:5$
. . The graph is above the xaxis.
We find that: .$\displaystyle f(2) \:=\:8  12  8 + 5 \:=\:7$
. . The graph is below the xaxis.
We find that:. $\displaystyle f(4) \:=\:64  48  16 + 5 \:=\:5$
. . The graph is above the xaxis.
So far, the graph looks like this: Code:

 (4,5)
(0,5)* *
 :
2  2 :
++++
:  : 4
:  :
:  :
*  *
(2,7)  (2,7}

Since a polynomial function is continuous, the graph must cross the xaxis
. . on the intervals $\displaystyle (2,0),\:(0,2).\:(2,4)$
And those are the three (real) zeros of the cubic function. Code:

* *
*  * :
2  2 :
+o+o+o+
:  : 4
:  : *
: *  * : *
*  *
