I would only like this tutor assigned to the assignment Randall Barnes Document Preview: (a) The polynomial function f is given by State the potential rational zeros of  QUOTE .Describe the end...

DISCRETE MATHEMATICS (MATH 225s)
1) (a) The polynomial function f is given by 153232)( 23  xxxxf
State the potential rational zeros of .
Rational zeros of the form


satisfy the following: p divides the
constant coefficient 15; q divides the leading coefficient 2.

The choices for p are: , and the choices for q are: .
Thus p/q is one of the following:





Now checking for all 12 values we get,
P(-5)=P(1/2)=(3)=0
Hence rational zeroes are -5, 1/2 and 3.
(i) Describe the end behaviour of and state the maximum number of turning
points that the graph of could have.
End behaviour -Since the degree of the polynomial, 3, is odd and the
leading coefficient, 2, is positive, then the graph of the given polynomial
falls to the left and rises to the right.
Turning points-
As the degree of polynomial is 3 , hence maximum no. of possible
turning points are 3-1=2
(ii) Show that x + 5 is a factor of
Calculating f(-5)

( ) ( ) ( ) ( )

( )
As f(-5)=0 hence from factor theorem x+5 is a factor of f(x).
(iii) Derive all the zeros of , state their respective multiplicities and whether
they touch or cross the x axis.
On factorizing

( )
( ) ( )( )( )

Now we set each factor to zero, we get
Zeroes are -5, 1/2 and 3.

Now in factorization, each factor only occurs once; therefore, the
multiplicity for each factor is 1. So the zeros -5, -3, and 3 each have a
multiplicity of 1.
As multiplicity is odd, hence the graph will cross x –axis at these zeroes.
(b) Three zeros of a polynomial function of degree four are 4, -1-2i and -7i.
Determine the polynomial f
As complex roots are in pairs always hence total zeros will be 4,-1-
2i,-1+2i,-7i, 7i.
And factors will be (x-4),(x-(-1-2i)),(X-(-1+2i)),(x-7i) and(x-(-7i))
Multiplying all
f(x)= (x-4)(x-(-1-2i))(X-(-1+2i))(x-7i)(x+7i)

( ) ( )( ( ))( ( ))( )( )

( ) ( )(( ) )( )
( ) ( )( )( )

( )

(c) Use appropriate transformations to sketch the graph of 2)3(
2
1
)( 6  xxf
Step I- Starting with graph of
Step II- Shifting graph 3 units left to get ( )
Step III- Stretching graph to get


( )
Step IV- Shifting graph 2 units up to get


( )
2) (a) Sketch the graph of the function given by
131)(  xxf . Identify all
asymptotes and intercepts if they exist. Also state the domain and range of the
function.
Using transformation –
Step...