Solutions to Remedial Exercises

© 2004 by Karl Hahn       Special thanks to Gordon Barrington for typesetting this page 		KCT logo (click for Home Page)

1) Find the sum of he following polynomials: 

   x3  +  3x2  -  4x  +  6
and 
   x4  -  7x3  -  6x2  +  x  +  7

Solution: 

1)  x4 - 6x3 - 3x2 - 3x + 13

2) Find the product of the following: 

    (x - 7) × (x2  -  5x  +  4) =  x3 - 12x2 + 39x - 28 





3) Use the distributive law to expand into polynomials: 
   a)  (x + 1)2 = x2 + 2x + 1

   b)  (x + 1)3 = x3 + 3x2 + 3x + 1

   c)  (x + 1)4 = x4 + 4x3 + 6x2 + 4x + 1








Now do the same for: 
   d)  (x - 1)2 = x2 - 2x + 1

   e)  (x - 1)3 = x3 - 3x2 + 3x - 1

   f)  (x - 1)4 = x4 - 4x3 + 6x2 - 4x + 1








4) Suppose that: 
   f(x) = x2  -  5x  +  4

and 
   g(x) = x  -  1

Write expanded polynomial expressions for the following: 
   f(x + 1) = (x+1)2 - 5(x+1) + 4
            = x2 + 2x + 1 - 5x - 5 + 4
            = x2 - 3x 


   f(x - y) = (x-y)2 - 5(x-y) + 4
            = x2 - 2xy + y2 - 5x + 5y + 4 


   f(x) + g(x) = x2 - 5x + 4 + x - 1
               = x2 - 4x + 3

   f(x) × g(x) = (x2 - 5x + 4)(x - 1)
               = x3 - 5x2 + 4x - x2 + 5x - 4
               = x3 - 6x2 + 9x - 4


   f(g(x)) = (x-1)2 - 5(x-1) + 4
           = x2 - 2x + 1 - 5x + 5 + 4
           = x2 - 7x + 10


   g(f(x)) = x2 - 5x + 4 - 1
           = x2 - 5x + 3


   f(g(x + 2) + 1) = f((x+1)+1)
                   = f(x+2)
                   = x2 + 4x + 4 - 5x - 10 + 4
                   = x2 - x - 2







5) The harmonic sum of x and y is given by: 

      (xy) / (x + y)







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