Leading coefficient test, Algebra

Assume that P ( x ) is a polynomial  along with degree n.  Thus we know that the polynomial have to look like,

                                                    P ( x ) =axn..............

We don't know if there are any other terms in the polynomial; however we know that the first term will need to be the one listed as it has degree n. Now we have the following facts regarding the graph of P ( x ) at the ends of the graph.

1.   If a> 0 and the value of n is even then the graph of P ( x )will increase without restricts positively at both endpoints.  A good instance of this is the graph of x2.

2369_Leading Coefficient Test.png

2.   If a= 0 and the value of n is odd then the graph of P ( x ) will increase without bound positively at the right end and decrease without bound at the left end.  A good instance of this is the graph of x3.

1749_Leading Coefficient Test1.png

3.   If a= 0 and the value of n is even then the graph of P (x ) will decrease without  any bound positively at both of the endpoints.  A good instance of this is the graph of -x2.

900_Leading Coefficient Test2.png

4.   If a= 0 and the value of n is odd then the graph of P ( x ) will decrease without  any bound positively at the right end and increase without  any bound at the left end.  A good instance of this is the graph of -x3.

990_Leading Coefficient Test3.png

Okay, now that we've obtained all that out of the way finally we can give a procedure for getting a rough sketch of the graph of a polynomial.

Posted Date: 4/8/2013 2:48:10 AM | Location : United States







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