Find a polynomial of degree 5 whose coefficients are real numbers and has the zeros -1, 4i and 1 – 2i. (5 marks) Use the Intermediate Value Theorem to determine whether the polynomial function has a...

1) Consider the polynomial function ( )
i) Determine:
a) The maximum number of zeros (1 mark)
= 4, because degree of polynomial = 4
b) The possible number of positive and negative zeros. (3 marks)
Number of positive roots = number of sign changes in f(x)
Sign changes: + + + - -
Sign changes = 1
Hence number of positive roots = 1
Number of negative roots = number of sign changes in f(-x)
( )

Sign changes: + - + + -
Sign changes = 3
Hence number of negative roots = 3
c) The list of potential rational zeros. (3 marks)

According to Rational Zero Theorem,
p/q is a rational zero where:
p = factor of constant term, a0
q = factor of leading coefficient, an

Here, a0 = -4
Factors are
an = 1
Factors are

Hence, potential rational zeros are:

d) The zeros of the polynomial and their multiplicities. (8 marks)

From the potential zeros, are zeros of the function as
f(1) =0 and f(-1) = 0 and f(-2) = 0.
Thus ( )( )( ) is a factor of the polynomial
( )( )( )
Using long division, we can find other roots.

Thus we get the root (x+2)
Root: +1, Multiplicity: 1
Root: -1, Multiplicity: 1
Root: -2, Multiplicity: 2
e) Whether each zero crosses of touches the x axis. (3 marks)

At x = 1 and x = -1, multiplicity = 1 = odd
Hence graph crosses x-axis
At x = 2, multiplicity = 2 = even
Hence graph touches x-axis.
f) The y intercept. (1 mark)
y-intercept is found when x = 0.
Put x = 0 in f(x)
y-intercept = -4
ii) Using the information in (i) above sketch the polynomial. (4 marks)
x y
1...