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A polynomial function f(x)f(x) with integer coefficients has a leading coefficient of 22 and a constant term of 7-7. According to the Rational Root Theorem, which of the following are possible roots of f(x)f(x)?\newlineMulti-select Choices:\newline(A) 1-1\newline(B) 12\frac{1}{2}\newline(C) 12-\frac{1}{2}\newline(D) 72-\frac{7}{2}

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Q. A polynomial function f(x)f(x) with integer coefficients has a leading coefficient of 22 and a constant term of 7-7. According to the Rational Root Theorem, which of the following are possible roots of f(x)f(x)?\newlineMulti-select Choices:\newline(A) 1-1\newline(B) 12\frac{1}{2}\newline(C) 12-\frac{1}{2}\newline(D) 72-\frac{7}{2}
  1. Understand Rational Root Theorem: The Rational Root Theorem states that any rational root, in the form of pq\frac{p}{q} (where pp is a factor of the constant term and qq is a factor of the leading coefficient), must be a factor of the constant term divided by a factor of the leading coefficient.
  2. List Factors of Constant Term: List all factors of the constant term 7-7: ±1\pm 1, ±7\pm 7.
  3. List Factors of Leading Coefficient: List all factors of the leading coefficient 22: ±1\pm1, ±2\pm2.
  4. Create Possible Fractions: Create all possible fractions p/qp/q using the factors of 7-7 for pp and the factors of 22 for qq: ±1/1\pm 1/1, ±7/1\pm 7/1, ±1/2\pm 1/2, ±7/2\pm 7/2.
  5. Simplify Fractions: Simplify the fractions: 1-1, 11, 7-7, 77, 12-\frac{1}{2}, 12\frac{1}{2}, 72-\frac{7}{2}, 72\frac{7}{2}.
  6. Match to Given Choices: Match the simplified fractions to the given choices: (A)1,(B)12,(C)12,(D)72(A) -1, (B) \frac{1}{2}, (C) -\frac{1}{2}, (D) -\frac{7}{2}.

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