Name                          .

 

Math 152B Midterm I

Please do all of the following problems.  Credit earned will be based on the steps that you show that lead to the final solution.  Good Luck!

 

Problem 1:  Factor the expression completely:

A)   x4 - 9x2  

Solution:  x2(x2 - 9)

 

 

 

 

 

 

 

 

 

 

 

 

B)    x2 + 3x - 54  

Solution:  (x + 9)(x - 6)

 

 

 

 

 

 

 

 

 

 

 

 

C)    6x2 + 11x - 10  

Solution: AC = -60, gives (15, -4)

            6x2 + 11x - 10 = 6x2 + 15x - 4x - 10   

            =  3x(2x + 5) - 2(2x + 5)  =  (3x - 2)(2x + 5)

 

 

 

 

 

 

 

 

 

 

 

 

D)    128xyz3 + 54x4y  

Solution:  2xy(64z3 + 27x3)  =  2xy(4z + 3x)(16z2 - 12xz + 9x2)

 

 

 

 

 

 

 

 

 

 

 

 

Problem 2:  Perform the indicated operations and express your answer in simplest form.

 

A       2x2 - x - 1     x2 - 6x + 8
                           .                                            
         x2 - 5x + 4    2x2 - 3x -2

              (2x + 1)(x - 1)     (x - 4)(x - 2)
   =                                .                               =   1                   
             (x - 4)(x - 1)       
(2x + 1)(x - 2)

 

 

 

 

 

 

 

 

 

 

 

 

B       x2 + 2 xy + y2           y + x
                                                                 
          x2 - x - 2         
       x - 2

                   x2 + 2xy + y2      x - 2
        =                              .                                    
                   x2 - x - 2         
    y + x

                   (x + y)(x + y)         x - 2
        =                               .                                    
                     
(x - 2)(x + 1)       x + y

                    (x + y)    
        =                                               
                    
(x + 1)  

 

 

 

 

 

 

 

 

 

 

 

 

 

Problem 3:  Solve the following equations

 

A.   x3 - 2x2 + x = 0  

Solution:  First factor

            x(x2 - 2x + 1) = 0

            x (x-1)2  =  0
    Now use the zero property 

            x = 0     or     x = 1

   

 

 

 

 

 

 

 

 

 

 

 

B.    4x2 = 15 - 4x

  Solution:  4x2 + 4x - 15 = 0

               AC  =  -60:   (10, -6)

        4x2 + 4x - 15  =  4x2 + 10x - 6x - 15

        =  2x(2x + 5) - 3(2x + 5)  =  (2x - 3)(2x + 5)  =  0

        2x - 3  =  0     or      2x + 5  =  0

        x = 3/2    or    x = -5/2

 

 

 

 

 

 

 

 

 

 

 

 

Problem 4:  Solve the following inequalities

 

A.     |4 - 3x|  < 2  

Solution:  4 - 3x  =  2    or     4 - 3x  =  -2

                -3x  =  -2    or    -3x  =  -6

                x  =  2/3    or    x  =  2

                [2/3, 2]

   

 

 

 

 

 

 

 

 

 

 

 

B.  |3x + 1| < 0  

 Solution:   3x + 1  =  0,    x  =  -1/3

   

 

 

 

 

 

 

 

 

 

 

 

 

 

Problem 5:  There is a lookout post situated in the center of a 3 mile circular trail.  How far is the lookout post from the trail?

 Solution:  

The circumference formula gives

                     C = 2pr   where 

                    C  =  the circumference (3 miles)

        and 

                    r = the radius (the distance from the lookout post to the trail)

                    3  =  2pr  

        divide by 2p  

                    r = 3/2p  

        The distance from the lookout post to the trail is 3/2p  miles.

 

 

 

 

 

 

 

 

 

 

 

                    

Problem 6:  Please answer the following true or false and explain your reasoning.

A.  If all terms of a trinomial are positive, all terms of both binomial factors will be positive

Solution:  True, (under the convention that the first coefficient if the first factor is always positive.)

 

 

 

 

 

 

 

 

 

 

 

 

 

B.  The solution to the inequality |x - 4|  <  -1 is the interval [3,5].

  Solution:  False,  The absolute value can never be negative, hence there is no solution.