Hello, Scotland!
We can use the Remainder/Factor Theorem.
. . \text{}1)^3  (\text{}1)^2  5(\text{}1)  3 \;=\;0" alt="f(\text{}1) \:=\\text{}1)^3  (\text{}1)^2  5(\text{}1)  3 \;=\;0" />
Since , then is a factor of
Then we have: .
b) One of the turning points of the graph of
lies on the xaxis. .Find the coordinates of this turning point. We can answer this without Calculus . . .
In part (a), we found that: . x+1)^2(x3)" alt="f(x) \:=\x+1)^2(x3)" />
The curve has xintercepts at .
The intercept has multiplicity 2.
. . Hence, the curve is tangent to the xaxis at
Therefore, there is a turning point at
The graph looks like this: Code:

 *


 *

o+o
* 1 *  * 3
* * *
*  *

* 