a.
To determine the degree of the polynomial we just need to add the powers of x, then we have:
[tex]3+2+1+1=7[/tex]Therefore, the degree of the polynomial is 7
b.
We know that any polynomial function can be writen as:
[tex]f(x)=(x-a_1)(x-a_2)(x-a_3)\cdots(x-a_{n-1})(x-a_n)[/tex]where ai are the zeros of the polynomial. The function given is already given in this form, then we have that the zeros of the function are:
[tex]\begin{gathered} x=-8165 \\ x=762 \\ x=574398 \\ x=-351 \end{gathered}[/tex]To determine the end behaviour of the function (when x is extremely large or extremely small) we just need to look at two things:
• If the larger power of x is even then the end behaviour will have the same sign of the leading coefficient; on the other hand, if the larger powers is odd then the end behaviour will have opposite sign to the leading coefficient.
,• The sign of the leading coefficient.
In this case the larger power of x is seven and the leading coefficient is negative. With this in mind we can answer the last two parts.
c.
In this case as x approaches minus infinity then the function will approach positive infinity.
d.
For this function, as x approaches infinity, then the function will approach minus infinity.
