Finding the roots of ( x 2 + 7 x + 6 ) 2 + 7 ( x 2 + 7 x + 6 ) + 6 = xI have been trying to solve this one problem from the Duke Math Meet, which does not provide a solution:Find all solutions of ( x 2 + 7 x + 6 ) 2 + 7 ( x 2 + 7 x + 6 ) + 6 = xAt first I tried to factorize the polynomial, but always had the right-hand side x remain, which was inconvenient. I then tried using the fact that one can write the left-hand side as a composition of functions, and equated that with the inverse of the quadratic plugged into the function, but that was a very nasty equation with square roots, still ending up with a quartic.What other solution paths are viable for this problem, and is there a way to factor the quartic?

Question

Finding the roots of   (      x    2    +  7  x  +  6      )    2    +  7  (      x    2    +  7  x  +  6  )  +  6  =  xI have been trying to solve this one problem from the Duke Math Meet, which does not provide a solution:Find all solutions of   (      x    2    +  7  x  +  6      )    2    +  7  (      x    2    +  7  x  +  6  )  +  6  =  xAt first I tried to factorize the polynomial, but always had the right-hand side x remain, which was inconvenient. I then tried using the fact that one can write the left-hand side as a composition of functions, and equated that with the inverse of the quadratic plugged into the function, but that was a very nasty equation with square roots, still ending up with a quartic.What other solution paths are viable for this problem, and is there a way to factor the quartic?

grizintimbp · Accepted Answer

Step 1Let  y  =      x    2    +  7  x  +  6  x  =      y    2    +  7  y  +  6thus  y  −  x  =  (  x  −  y  )  (  x  +  y  )  +  7  (  x  −  y  )so either  x  =  yor  −  1  =  x  +  y  +  7Both of these cases reduce the problem to simple quadratic equations.And you get  x  =  −  4  ±      2    ,  −  3  ±      3

Nicholas Hunter · Accepted Answer

Step 1Define   f  (  x  )                    =                    def                  x    2    +  7  x  +  6. Then, as you&#039;ve noted, this equation equivalent to finding the roots to the quartic  h  (  x  )  =  f  (  f  (  x  )  )  −  x  .Write   f  [  x  ,  y  ]  =  x  +  y  +  7 for the divided difference of f, i.e., the unique bivariate polynomial such that  f  (  x  )  −  f  (  y  )  =  (  x  −  y  )  f  [  x  ,  y  ]  ,as calculated via the Division Algorithm.Step 2Then the trick is to add and subtract f(x) and recognize the divided difference:                    h        (        x        )                            =        f        (        f        (        x        )        )        −        x                                          =        f        (        f        (        x        )        )        −        f        (        x        )        +        f        (        x        )        −        x                                          =        (        f        (        x        )        −        x        )        f        [        f        (        x        )        ,        x        ]        +        f        (        x        )        −        x                                          =        k        (        x        )        g        (        x        )            where k(x) and g(x) are quadratic polynomials                    k        (        x        )                                                                =                                      def                                      f        (        x        )        −        x                                  =                  x          2                +        6        x        +        6                                  =        (        x        +        3                  )          2                −        3                            g        (        x        )                                                                =                                      def                                      f        [        f        (        x        )        ,        x        ]        +        1                                  =                  x          2                +        8        x        +        14                                  =        (        x        +        4                  )          2                −        2        .

Finding the roots of (x^2+7x+6)^2+7(x^2+7x+6)+6=x

Answered question

Answer & Explanation

New Questions in Algebra I