Determine whether -2 and 10 are upper or lower bounds of x^4–x^3-11x^2+9x+18 = 0. Use synthetic division.Show step by step solution to explain the answer.

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embizze | High School Teacher | (Level 1) Educator Emeritus

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A number k is a lower bound of potential roots if, when using synthetic division with k as the divisor, the coefficients of teh quotient and remainder alternate in sign.

(1) -2 is not a lower bound: Perform synthetic division with -2 as the divisor:

-2  | 1  -1  -11  9    18
           -2     6  10 -38
      -------------------
       1   -3   -5   19 -20

** Since the coefficients do not alternate in sign, -2 is not a lower bound.**

-- -4 is a lower bound:

-4  | 1  -1   -11  9    18
           -4    20  -36 108
       --------------------
       1   -5   9   -27   126

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A number k is an upper bound for potential roots if , after divinding using synthetic division with k as the divisor, the coefficients of the quotient and remainder are all positive.

(2) 10 is an upper bound for roots: Use synthetic division

10 | 1  -1  -11   9    18
           10  90  790 7990
     ----------------------
      1  9   79   799   8008

** Thus 10 is an upper bound for the possible roots.**

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`x^4-x^3-11x^2+9x+18=(x+3)(x+1)(x-2)(x-3)` so the roots are all rational and are -3,-1,2,3.

Sources:

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