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)  

 

 C        2             x
                   +                              
         5 - x       x - 5

Solution

First factor a -1 out of the first denominator in order to put it in standard form

          -2            x
                  +                              
         x - 5       x - 5

Now we have a common denominator.  Just add

          -2 + x    
    =                                   
            x - 5    

            x - 2    
    =                                   
            x - 5    

 

 

 D        x                  3
                     +                                      
         x2 - 1       x2 + 4x + 5

Solution

First factor the denominators

                     x                                3
     =                               +                                               
            (x - 1)(x + 1)              (x + 5)(x + 1)

The LCD is (x - 1)(x + 1)(x + 5).  Build the two fractions so that they have the same denominator.

                  x (x + 5)                                 3(x - 1)
     =                                            +                                                        
            (x - 1)(x + 1)(x + 5)                (x - 1)(x + 5)(x + 1)

Now add the numerators

                x2 + 5x + 3x - 3    
     =                                                           
            (x - 1)(x + 1)(x + 5)           

                x2 + 8x - 3    
     =                                                           
            (x - 1)(x + 1)(x + 5)           

Since the numerator does not factor, we are done.

 

 

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: 

The pressure p in pounds per square foot of a wind is directly proportional to the square of the velocity v of he wind.  If a 10-mi/hr wind produces a pressure of 0.3 lb/ft2, what pressure will a 100-mi/hr wind produce?

Solution

        This is a variation problem.  The first sentence implies

        p  =  kv2

        The second sentence tells us

        0.3  =  k(10)2        When v  =  10  p  =  0.3

        0.3  =  100k

        k  =  0.3/100  =  0.003        Dividing both sides by 100

        p  =  0.003v2        Substituting k  =  0.003 back into the original equation

        Now we want to know what p is when v  =  100

        p  =  0.003(100)2        Substituting 100 in for v

        p  =  0.003(10,000)  =  30

        A 100 mile per hour wind will produce a pressure of 30lb/ft2 

 

Problem 5: 

Simplify the following rational expression

          x3 - 8    
                                         
          x2 - 4   

 Solution

Notice that the numerator is a difference of cubes and the denominator is a difference of squares.  We factor

            (x - 2)(x2 + 2x + 4)    
     =                                                       
               (x - 2)(x + 2)   

Now cancel to get

            (x2 + 2x + 4)    
   =                                               
                (x + 2)