# SOLUTION: Use the rational roots therm to list all the possible rational roots for the equation then find all the roots : f(x) x^3-2x^2-17x+10

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 Question 410194: Use the rational roots therm to list all the possible rational roots for the equation then find all the roots : f(x) x^3-2x^2-17x+10Answer by Edwin McCravy(8908)   (Show Source): You can put this solution on YOUR website!Use the rational roots therm to list all the possible rational roots for the equation then find all the roots : f(x) = x³ - 2x² - 17x + 10 ``` All the possible rational roots have numerators which are divisors of 10, positive or negative, and whose denominators are the divisors of the coefficient of x³, which is 1. Since 1 has only 1 divisor, 1, the only possible roots are ± the divisors of 10. ±1, ±2, ±5, ±10 Try 1 1|1 -2 -17 10 | 1 -1 -18 1 -1 -18 -8 No, 1 is not a root because we get -8 on the bottom right as a remainder, not 0. -1|1 -2 -17 10 | -1 3 14 1 -3 -14 24 No, -1 is not a root because we get 24 on the bottom right as a remainder, not 0. 2|1 -2 -17 10 | 2 0 -34 1 0 -17 -24 No, 2 is not a root because we get -24 on the bottom right as a remainder, not 0. -2|1 -2 -17 10 | -2 8 18 1 -4 -9 28 No, -2 is not a root because we get 28 on the bottom right as a remainder, not 0. 5|1 -2 -17 10 | 5 15 -10 1 3 -2 0 Yes!! Finally!! 5 is a root because we get 0 on the bottom right. So we have now factored the polynomial f(x) = x³ - 2x² - 17x + 10 as f(x) = (x - 5)(x² + 3x - 2) The trinomial in the parentheses does not factor, so we set each factor = 0 Setting the first factor = 0: x - 5 = 0 x = 5 Setting the second factor, the trinomial, = 0: x² + 3x - 2 = 0 We use the quadratic formula: a = 1, b = 3, c = -2 So the three roots are 5, , Edwin```