Answer
a. $x=\frac{1}{3}, \frac{-1\pm i\sqrt 3}{2}$
b. See graph and explanations.
Work Step by Step
a. Based on the graph, we can identify a zero as $x=\frac{1}{3}$. Using synthetic division, we can find the quotient as $3x^2+3x+3=3(x^2+x+1)$, as shown in the figure. Thus the zeros are $x=\frac{1}{3}, \frac{-1\pm i\sqrt 3}{2}$
b. The end behaviors of the function can be found as $x\to-\infty, y\to-\infty$ and $x\to\infty, y\to\infty$. The maximum number of turning points is $3-1=2$ (since there is only one real zero, the function has no turning point). The y-intercept can be found as $y=f(0)=-1$. With the above information, we can finish the graph as shown in the figure.
