SOLUTION: If possible, sketch a graph of a polynomial that satisfies the conditions. Let a be the leading coefficient. Degree 5 and symmetric with respect to the y-axis.

1. It is IMPOSSIBLE to have a polynomial of ODD degree 
greater than 1 which is symmetrical with respect to the 
Y-AXIS.  

So the answer to your question is "It is impossible". 

2. It is POSSIBLE to have a polynomial of EVEN degree 
which is symmetrical with respect to the Y-AXIS.

3. It is POSSIBLE to have a polynomial of ODD degree 
greater than 1 which is symmetrical with respect to the 
ORIGIN.  

4. It is IMPOSSIBLE to have a polynomial of EVEN degree 
which is symmetrical with respect to the ORIGIN.

The reason for the above is given by the extreme right 
and left hand behavior rules, which are given below:

Leading Coefficient Test (right hand behavior)
•If the leading coefficient, an, of the polynomial is 
positive, then the right hand side of the graph will 
rise towards + infinity. 
•If the leading coefficient, an, of the polynomial is 
negative, then the right hand side of the graph will fall
towards - infinity.
Degree of the Polynomial (left hand behavior)
•If the degree, n, of the polynomial is even, the left hand side 
will do the same as the right hand side.
•If the degree, n, of the polynomial is odd, the left hand side 
will do the opposite of the right hand side.

In order to be symmetrical with respect to the y-axis, a 
polynomial must do the same on both extreme sides, so its degree
must be even.

In order to be symmetrical with the origin, a polynomial must do
the opposite on both extreme sides, so its degree must be odd.


Edwin