Answer
* Notice that the polynomial has a range $[-\infty, 20]$.
* 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 five 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 6, although it could also be of degree 8, 10, etc.
* Finally, notice also that the polynomial becomes a large negative number as $x$ gets large in magnitude, either positive or negative, so the leading coefficient must be negative sign.
Work Step by Step
* Notice that the polynomial has a range $[-\infty, 20]$.
* 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 five 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 6, although it could also be of degree 8, 10, etc.
* Finally, notice also that the polynomial becomes a large negative number as $x$ gets large in magnitude, either positive or negative, so the leading coefficient must be negative sign.