To determine the number of possible rational roots of a polynomial, we can use the Rational Root Theorem. This theorem states that any rational root, expressed as a fraction \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. For the polynomial \( x^4 + 2x^3 - 7x^2 - 8x + 12 \), the constant term is 12 and the leading coefficient is 1. We will find all possible combinations of these factors to determine the number of possible rational roots.
The given polynomial is \( x^4 + 2x^3 - 7x^2 - 8x + 12 \). The constant term is 12, and the leading coefficient is 1.
According to the Rational Root Theorem, any rational root \( \frac{p}{q} \) must have \( p \) as a factor of the constant term (12) and \( q \) as a factor of the leading coefficient (1).
- Factors of 12 (possible values for \( p \)): \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)
- Factors of 1 (possible values for \( q \)): \( \pm 1 \)
Using the factors identified, the possible rational roots are:
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12
\]
The set of possible rational roots is:
\[
\{1.0, 2.0, 3.0, 4.0, 6.0, 12.0, -1.0, -2.0, -3.0, -4.0, -6.0, -12.0\}
\]
This set contains 12 distinct values.
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