Answer
* This must be a polynomial of even degree, because if the highest power
of $x$ is an odd power, the polynomial can take on all real numbers,
positive and negative.
* Notice that it has three turning points. Observe that a polynomial of degree $n$ has at most $n-1$ turning points. So the polynomial graphed in the Figure might be degree 4, although it could also be of degree 6, 8, etc.
* Finally, notice also that the polynomial becomes a large positive number as $x$ gets large in magnitude, either positive or negative, so the leading coefficient must be positive sign.
Work Step by Step
* Notice that the polynomial has a range $[-2, \infty]$.
* This must be a polynomial of even degree, because if the highest power
of $x$ is an odd power, the polynomial can take on all real numbers,
positive and negative.
* Notice that it has three turning points. Observe that a polynomial of degree $n$ has at most $n-1$ turning points. So the polynomial graphed in the Figure might be degree 4, although it could also be of degree 6, 8, etc.
* Finally, notice also that the polynomial becomes a large positive number as $x$ gets large in magnitude, either positive or negative, so the leading coefficient must be positive sign.
