1. Fill in the following table of values:

x	-4	-3	-2	-1	0	1	2
f(x)

2. On which of the following intervals does the Intermediate Value Theorem say that f(x) has a real zero?

 
• between -4 and -3
   
• between -3 and -2
   
• between -2 and -1
   
• between -1 and 0
   
• between 0 and 1
   
• between 1 and 2
   
• between 2 and 3
   
3.   (Optional)   If you have a graphing calculator, plot the graph of  f.
    

2.   Compute the following complex numbers:

1. (3 + 4i )·(4 + i)
    
2. (3 + 2i )·(2 - i)
    
3. the complex conjugate of 4 + 3i
    
4. the complex conjugate of - 6i
    
5. 
    
6. 
    

3.  Use the method of exercise 10.2.7 to approximate .  Start with  a=3,  and stop when you get a rational number  c  such that  c² - 10  is less than  .00000001 = 10^-8.
 

4.   Find all of the zeros of the polynomial:

              x³ + 2x² - x - 2

Factor this polynomial as a product of 3 linear factors.
  

5.   Find a rational zero of each polynomial, and then factor the polynomial as a product of a linear factor and a quadratic factor.

1. 2x³ - 7x² + 5x - 1
    
2. x³ - 7x² + 5x + 1     (Please note the correction.)

 

6.   For each polynomial in the preceding problem, find all of its zeros (whether rational, real, or complex).
Suggestion: Use the quadratic formula to find the zeros of the quadratic factor.
 

1. Explain why   and   are algebraic.
2. Write  ( + )²  as  A + B, where  A and  B are rational numbers.
3. Write  ( + )⁴  as  A + B, where  A and  B are rational numbers.
4. Find rational numbers  S and  T such that ( + )⁴  + S( + )²  + T = 0.
   (This will show that  +  is also algebraic.)
   Suggestion: Use the answers from the previous parts of this exercise to set up a system of equations.
   (But if you see certain patterns, you may be able to first figure out what  S has to be, and then determine the value of  T.) 
    

8.  Notation: If  z is a complex number, we'll write to denote its complex conjugate. Thus, if  z = a + bi, then = a - bi. So, if  z and  w are complex numbers, the point of this exercise is to compare the product of the conjugates  with the conjugate of the product .

• In each case, calculate the product  zw and the conjugate of the product.
   
1. z= 2+3i and  w= 4 + i     zw = ___________________
    
        = _________________
    
2. z= 3+2i and  w= 1 - 2i     zw = ___________________
    
        = _________________
    
3. z = a + bi and  w = c + di     zw = ____________________
    
         = _________________
    
• In each case, calculate the conjugates and and then the product of the conjugates.
    
1. z= 2+3i and  w= 4 + i     = _______     = _______
    
        = _________________
    
2. z= 3+2i and  w= 1 - 2i     = _______     = _______
    
        = _________________
    
3. z== a + bi and  w = c + di     = _______     = _______
    
         = _________________
    
• What is the relation between the conjugate of the product and the product of the conjugates??
   

9.   Review the exercises that were done as group work and homework assignments
 

Click here to see some of the solutions.