SOLUTION: Use the rational roots theorem to solve: p(x)=x^4-5x^3+5x^2+5x-6

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 Click here to see ALL problems on Graphs Question 148851: Use the rational roots theorem to solve: p(x)=x^4-5x^3+5x^2+5x-6 Answer by Edwin McCravy(13211)   (Show Source): You can put this solution on YOUR website! ``` Use the rational roots theorem to solve: The last term is -6, which in absolute value is 6, and which has these factors 1,2,3,6 The leading term (the term with largest exponent) is , has coefficient 1, which in absolute value is 1, and which has only the one factor 1. Now we form all the fractions with numerator 1,2,3,or 6 and denominator 1 These are , , , or , , , . Their negatives are also possible rational roots, so all the possible rational roots are: ±, ±, ±, ± We start out by trying using synthetic division to see if we get a 0 remainder: 1| 1 -5 5 5 -6 | 1 -4 1 6 1 -4 1 6 0 Yes we do get 0 remainder, so we know that we have factored the polynomial as So now we can just find the roots of the simpler polynomial: The first and last numbers happen to be the same as they were in the original, so we can try the same ones again. We try 1 again: 1| 1 -4 1 6 | 1 -3 -2 1 -3 -2 4 No that leaves a remainder of 4, not 0. So we try -1 -1| 1 -4 1 6 | -1 6 -6 1 -5 6 0 Yes we do get 0 remainder, so we know that we have factored the polynomial again. First we factored as Now we have factored the polynomial in the second parentheses, and we have: So now we can just find the roots of the simpler polynomial: But we don't need to do synthetic division again, for factors as So now we have factored completely: Set each factor equal to 0 and so the roots are , , , Edwin```