The first feature of a polynomial function is that its graph is continuous, that is, the graph of a polynomial function has no breaks. The second feature is that the graph of a polynomial function has only smooth, rounded turns.
Since the problem establishes that we may create a sample polynomial to be used in our explanations, we can say that this polynomial function is:
[tex]R(x)=-2x^4+2x^2[/tex]
To find the x-intercepts of R(x) set R(x) equal to zero and solve for x, so:
[tex] Set \ R(x) \ equal \ to \ zero: \\ \\ -2x^4+2x^2=0 \\ \\ Remove \ common \ monomial \ factor: \\ \\ -2x^2(x^2-1)=0 \\ \\ Factor \ out: \\ \\ -2x^2(x-1)(x+1)=0 [/tex]
So the real zeros are:
[tex] \boxed{x=0} \\ \\ \boxed{x=1} \\ \\ \boxed{x=-1} [/tex]
To construct a rough graph of R(x) we start setting the zeros on the x-axis. On the other hand, given that the leading coefficient [tex]a_{n}=-2[/tex] is negative, that is, [tex]a_{n}<0[/tex]. The graph falls to the left and right. So the graph is shown in the Figure below.
