A polynomial function f(x) with integer coefficients has a leading coefficient of 7 and a constant term of 5. According to the Rational Root Theorem, which of the following are possible roots of f(x)? Multi-select Choices: (A)-10/17 (B)-5/9 (C)-8 (D)-1

Question

A polynomial function f(x) with integer coefficients has a leading coefficient of 7 and a constant term of 5. According to the Rational Root Theorem, which of the following are possible roots of f(x)?   Multi-select Choices: (A)-10/17 (B)-5/9 (C)-8 (D)-1

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Theorem Explanation: The Rational Root Theorem states that any rational root, in the form of p/q (where p and q are integers), of a polynomial equation with integer coefficients must have p as a factor of the constant term and q as a factor of the leading coefficient.Constant Term Factors: List the factors of the constant term 5: ±1, ±5.Leading Coefficient Factors: List the factors of the leading coefficient 7: ±1, ±7.Generate Possible Roots: Generate the possible rational roots by taking all combinations of factors of the constant term over factors of the leading coefficient: ±1/1, ±5/1, ±1/7, ±5/7.Simplify Roots List: Simplify the list of possible rational roots: ±1, ±5, ±1/7, ±5/7.Check Given Choices: Check which of the given choices match the possible rational roots: (A) -10/17 is not a possible root because 17 is not a factor of 7. (B) -5/9 is not a possible root because 9 is not a factor of 7. (C) -8 is not a possible root because 8 is not a factor of 5. (D) -1 is a possible root because it is in the list of possible rational roots.

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