# 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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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

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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

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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

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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.