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.