MI 3
March 9, 2012
2 pts.
Poly Unit Test
Name: ______________
Show all work! Calculators are not allowed.
State whether the following statements are true or false. Briefly explain your reasoning.
False 1) y  2( x  5)2  3 has two real zeros.
y  0 for all x since ( x  5)2  0 for all x . Therefore this quadratic has no real roots.
2 pts.
False 2) If a polynomial with real coefficients has a root of x  2  i , then ( x  (2  i )) is a factor
of the polynomial.
The factors would be ( x  (2  i )) and ( x  (2  i ))
2 pts.
False 3) The graph of the polynomial y  x5  3x3  6 x  5 could be as shown below:
This is the graph of an even degree polynomial since y  , for x   and x  
4) Write the equation of a seventh degree polynomial, with integer coefficients, with a bounce
point at x  3 , a pass- through point at x  6 , and an imaginary root at 1  i . Write out your
polynomial equation in factored form.
One possibility is: f ( x)  ( x  3)4 ( x  6)( x  (1  i))( x  (1  i))
MI-3
v1
SP ‘12
2 each
5) Determine whether each of the following is a polynomial. If it is a polynomial,
state its degree, if not a polynomial, explain why.
3
g ( x)  6 x 2  2  2 Not a polynomial since the last term has x raised to a negative
x
2
power, 3x .
k ( x)  3 x8  5 x 7  1
Is a polynomial of degree 8.
3 pts.
6) Evaluate (5  3i)  (2  i) .
(5  3i)  (2  i)  (3  4i)  32  42  5
4 pts.
7) Solve for z and express the answer in a  bi form:
(3  2i ) z  (4  i)  2  3i  (3  2i) z  2  4i
2  4i
z
(3  2i )
(2  4i ) (3  2i)
z

(3  2i ) (3  2i )
(14  8i)
z
13
2 pts.
8) Find the sum and the product of the roots for the given polynomial without actually finding
its roots:
g ( x)  2 x5  8 x 4  5 x 2  14
Sum = 
8
 4
2
Product = 
14
 7
2
9) Write the equation of a polynomial function of smallest degree with the graph given below.
y  a( x  1)3 ( x  2)2 ( x  3) . Since
y(0)  48  a 1 4  (3), so a  4  y  4( x  1)3 ( x  2)2 ( x  3)
48

1
10) Sketch the graph of p ( x) 
3 pts.
2
3
1
( x  1) 2 ( x  2)3 ( x  4) . Indicate x and y intercepts clearly.
2
11) Given f ( x)  ( x  1)( x  2)( x  1) . Find c so that 2 is a zero of y  f ( x)  c .
If 2 is a root of y  f ( x)  c , then 0  f (2)  c  0  (2  1)(2  2)(2  1)  c  c  12
4 pts.
f  x   2x3  3x2  4x  15
12) Determine the exact values, over the
complex numbers, of all the zeros of :
y
f  x   2x3  3x2  4x  15


Show your work clearly.




x










4 pts.
13) Solve z 3  27  0 and graph and label the solutions clearly.
z
im aginary
real
