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Find the sum of all real numbers $x$ that are not in the domain of the function $$f(x) = \frac{1}{x^2+7} + \frac{1}{x^3 - x^4} + \frac{1}{x^2 - 3x + 2}.$$

 Jan 21, 2024
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         To not be in the domain of a function, then x must be a value in which the denominator is equal to 0 and thus the expression is undefined. x^2 + 7 can never be 0 for a real value of x, because x^2 is nonnegative. x^3 - x^4 = x^3(1 - x), which this expression is 0 when x = 0 and 1. Additionally, x^2 - 3x + 2 = (x - 2)(x - 1) is 0 for when x = 2 and 1.

         Altogether, we have x = 0, 1, or 2 that will result in one of the fractions having a denominator of 0... Since 1/0 is undefined, x = 0, 1, and 2 are not part of the domain of f.

         Hence, their sum 0 + 1 + 2 = 3, which is your final answer.

 Feb 19, 2024

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