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given that (x-1) and (x+1) are factors of px^3 + qx^2 - 3x - 7 find the values of p and q.

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3 Answers

If we multiply (x-1)(x+1) = (x2-1) we now have to find what factor when multiplied by (x2-1) will yield px3+qx2-3x-7.  we know that the factor will have +7 as the only way that we can get -7 will be multiplying (-1)7, and the same way we know that in order to get px3 is if we have x2*px.  Now let is build the factor and test the equation.
 
(x2-1)(px+7) = px3 +7x2-px-7,  and we look at the original equation px3+qx2- 3x - 7 and we can conclude that q=7 and p=3 .

Comments

p X^3 + q X^2 - 3X - 7
 
 Has a factor of ( X -1) , then X =1 is the root of the polynomial
 
 
    p ( 1) ^3 + q ( 1) ^2 - 3 ( 1) - 7 =0
 
        P + q = 10     ( 1)
 
       Factor of  ( X +1) , means that X = -1 is the root of the polynomial:
 
        p ( -1) ^3 + q(-1)^2 - 3 ( -1) -7 =0
      
          -p + q = 4      ( 2)
      
      From equation ( 1) & ( 2) :
 
          2q = 14      q= 7
 
           P + 7 = 10         p = 3
 
  Test Polynomial :
 
     3 X^3 + 7 X^2 - 3x  -7
 
     3 X^2 - 3X + 7 X^2 -7
 
     3 X^3- 3X  + 7 X^2 -7
 
     3x ( X^2 - 1) + 7 ( X^2 -1) =
 
       ( X +1) ( X-1) ( 3X +7 ) = 0
 
        Zeros : X = -1  , X = 1  , X = -7/3

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