document.write( "Question 72593: I need someone really smart.\r
\n" ); document.write( "\n" ); document.write( "Let f(x) =x3 -8x2 +17x -9. use the factor theorem to find other solutions to f(x) -f(1) = 0, beside x=1.\r
\n" ); document.write( "\n" ); document.write( "I have 6 algebra books that can not explain this factor theorem whereas I can understand it, Can someone out there explain an easy way to understand this.\r
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Algebra.Com's Answer #52026 by rmromero(383) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "Let f(x) =x3 -8x2 +17x -9. use the factor theorem to find other solutions to f(x) -f(1) = 0, beside x=1.\r
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document.write( "First let us evaluate f(x) =x3 -8x2 +17x -9, given f(1)\r
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document.write( "Using Factor theorem:\r
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document.write( "    f(1) = x3 -8x2 +17x -9
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document.write( "         = 1^3 - 8(1)^2 + 17(1) - 9
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document.write( "         = 1 - 8 + 17 - 9
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document.write( "         = 18 - 17
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document.write( "    f(1) = 1\r
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document.write( "Condition: f(x) -f(1) = 0 , 
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document.write( "               1 - 1 = 0   right?
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document.write( "When f(x) = x3 -8x2 +17x -9 , f(x) = 1,  f(x) -f(1) = 0
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document.write( " f(x) = x3 -8x2 +17x -9
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document.write( "   1 = x3 -8x2 +17x -9
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document.write( "   0 = x3 -8x2 +17x -10   <<<<<< Find roots of the polynomial\r
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document.write( "Any factor of f(x) = x3 -8x2 +17x -10  will make f(x) -f(1) = 0.Let us find 
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document.write( "possible factors/roots of f(x) = x3 -8x2 +17x -10.\r
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document.write( " Possible roots = Factor of the constant 10 over coefficient x^3 which is 1.
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document.write( " Factors of 10 = 1, -1, 2, -2, 5, -5, 10, -10
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document.write( " Factors of 1 = 1, -1\r
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document.write( " Possible roots are: {1, -1, 2, -2, 5, -5, 10, -10}\r
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document.write( "         Use Factor theorem to test which of the following are 
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document.write( "                  the root of the polynomial\r
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document.write( "f(1) = 1^3 -8(1)^2 +17(1) -10   
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document.write( "     = 1 - 8 + 17 - 10
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document.write( "     = 18 - 18
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document.write( "     = 0    ----------> This is one root of the polynomial\r
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document.write( "f(-1) = (-1)^3 -8(-1)^2 +17(-1) -10   
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document.write( "      = -1 - 8 - 17 - 10
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document.write( "      = -36  ---------> since this has remainder, this is not a root of the polynomial\r
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document.write( "F(2) = 2^3 -8(2)^2 +17(2) -10   
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document.write( "     = 8 - 32 + 34 - 10
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document.write( "     = 42 - 42
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document.write( "     = 0  --------> Root of the polynomial\r
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document.write( "f(-2) = (-2)^3 -8(-2)^2 +17(-2) -10   
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document.write( "      = -8 - 32 - 34 - 10
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document.write( "     = -84  ----------> not a root\r
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document.write( "f(5) = 5^3 -8(5)^2 +17(5) -10   
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document.write( "     = 125 - 200 + 85 - 10
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document.write( "     = 210 - 210
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document.write( "     = 0  ----------> root of the polynomial\r
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document.write( "f(-5) = (-5)^3 -8(-5)^2 +17(-5) -10  
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document.write( "      = -125 - 200 - 85 - 10
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document.write( "      = - 420 ----> not a root\r
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document.write( "f(10) = 10^3 -8(10)^2 +17(10) -10    
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document.write( "      = 1000 - 800 + 170 - 10
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document.write( "      = 1170 - 810
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document.write( "      = 360 ----------> not a root\r
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document.write( "f(-10) = (-10)^3 -8(-10)^2 +17(-10) -10  
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document.write( "       = -1000 - 800 - 170 - 10
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document.write( "       = -1980 -------------> not a root\r
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document.write( "Therefore aside from x = 1, the other solutions are x = 2 and x = 5\r
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document.write( "Checking:\r
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document.write( "f(x) =x3 -8x2 +17x -9, f(x) - f(1) = 0\r
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document.write( "when x = 2
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document.write( "      f(2) - f(1) = 0
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document.write( "      (2^3 -8(2)^2 +17(2)-9) - (1^3 -8(1)^2 +17(1) -9)= 0
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document.write( "                   8 - 32 + 34 - 9 - (1 - 8 + 17 - 9) = 0
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document.write( "                                           42 - 41 -1 = 0
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document.write( "                                                    0 = 0 \r
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document.write( "When x = 5 
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document.write( "     f(5) - f(1) = 0
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document.write( "     5^3 -8(5)^2 +17(5)- 9 - (1^3 -8(1)^2 +17(1) -9) = 0
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document.write( "               125 - 200 + 85 - 9 - (1 - 8 + 17 - 9) = 0
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document.write( "                                       210 - 209 - 1 = 0
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document.write( "                                                   0 = 0      \r
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document.write( "If you have questions, just ask ok.\r
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