Question 72593
<pre>

Let f(x) =x3 -8x2 +17x -9. use the factor theorem to find other solutions to f(x) -f(1) = 0, beside x=1.

First let us evaluate f(x) =x3 -8x2 +17x -9, given f(1)

Using Factor theorem:

    f(1) = x3 -8x2 +17x -9
         = 1^3 - 8(1)^2 + 17(1) - 9
         = 1 - 8 + 17 - 9
         = 18 - 17
    f(1) = 1


Condition: f(x) -f(1) = 0 , 
               1 - 1 = 0   right?
When f(x) = x3 -8x2 +17x -9 , f(x) = 1,  f(x) -f(1) = 0
 f(x) = x3 -8x2 +17x -9
   1 = x3 -8x2 +17x -9
   0 = x3 -8x2 +17x -10   <<<<<< Find roots of the polynomial

Any factor of f(x) = x3 -8x2 +17x -10  will make f(x) -f(1) = 0.Let us find 
possible factors/roots of f(x) = x3 -8x2 +17x -10.

 Possible roots = Factor of the constant 10 over coefficient x^3 which is 1.
 Factors of 10 = 1, -1, 2, -2, 5, -5, 10, -10
 Factors of 1 = 1, -1

 Possible roots are: {1, -1, 2, -2, 5, -5, 10, -10}

         Use Factor theorem to test which of the following are 
                  the root of the polynomial

f(1) = 1^3 -8(1)^2 +17(1) -10   
     = 1 - 8 + 17 - 10
     = 18 - 18
     = 0    ----------> This is one root of the polynomial

f(-1) = (-1)^3 -8(-1)^2 +17(-1) -10   
      = -1 - 8 - 17 - 10
      = -36  ---------> since this has remainder, this is not a root of the polynomial

F(2) = 2^3 -8(2)^2 +17(2) -10   
     = 8 - 32 + 34 - 10
     = 42 - 42
     = 0  --------> Root of the polynomial

f(-2) = (-2)^3 -8(-2)^2 +17(-2) -10   
      = -8 - 32 - 34 - 10
     = -84  ----------> not a root

f(5) = 5^3 -8(5)^2 +17(5) -10   
     = 125 - 200 + 85 - 10
     = 210 - 210
     = 0  ----------> root of the polynomial

f(-5) = (-5)^3 -8(-5)^2 +17(-5) -10  
      = -125 - 200 - 85 - 10
      = - 420 ----> not a root

f(10) = 10^3 -8(10)^2 +17(10) -10    
      = 1000 - 800 + 170 - 10
      = 1170 - 810
      = 360 ----------> not a root

f(-10) = (-10)^3 -8(-10)^2 +17(-10) -10  
       = -1000 - 800 - 170 - 10
       = -1980 -------------> not a root

Therefore aside from x = 1, the other solutions are x = 2 and x = 5


Checking:

f(x) =x3 -8x2 +17x -9, f(x) - f(1) = 0

when x = 2
      f(2) - f(1) = 0
      (2^3 -8(2)^2 +17(2)-9) - (1^3 -8(1)^2 +17(1) -9)= 0
                   8 - 32 + 34 - 9 - (1 - 8 + 17 - 9) = 0
                                           42 - 41 -1 = 0
                                                    0 = 0 

When x = 5 
     f(5) - f(1) = 0
     5^3 -8(5)^2 +17(5)- 9 - (1^3 -8(1)^2 +17(1) -9) = 0
               125 - 200 + 85 - 9 - (1 - 8 + 17 - 9) = 0
                                       210 - 209 - 1 = 0
                                                   0 = 0      



If you have questions, just ask ok.