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The Rational Root Theorem is a useful tool for finding the zeros of a polynomial. It gives us a systematic way to list possible rational roots of a polynomial equation.
Here's how it works:

• First, identify the constant term and the leading coefficient of your polynomial. In the example polynomial, \( P(x) = 2x^3 - 8x^2 + 9x - 9 \), the constant term is \(-9\) and the leading coefficient is \(2\).
• The possible rational roots are fractions \( \frac{p}{q} \), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.
• For \(-9\), the factors are \(\pm 1, \pm 3, \pm 9\), and for \(2\), they are \(\pm 1, \pm 2\).
• Thus, the potential rational roots are \(\pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2}\).

This method narrows down potential candidates for the polynomial's zeros, allowing us to check each one until we find an actual root that makes the polynomial equal to zero.

Once a possible zero of a polynomial is identified, polynomial division becomes the next logical step to simplify the polynomial. Suppose we find \( x = 3 \) as a root for our polynomial \( P(x) \). Polynomial division helps us divide \( P(x) \) by \( x - 3 \).There are two common methods:

• Long Division: Similar to numerical long division, align terms by decreasing powers and divide step by step.
• Synthetic Division: Requires less writing space and handles the coefficients directly, offering a quicker computation for polynomials.

Through either method, dividing \( P(x) = 2x^3 - 8x^2 + 9x - 9 \) by \( x - 3 \) results in the quotient \( 2x^2 - 2x + 3 \). This new polynomial is of a lower degree, making it easier to solve and continue the search for the remaining zeros.

When solving for the zeros of a polynomial, finding a negative discriminant indicates the presence of complex solutions. In the example, after using polynomial division, we have the quadratic \( 2x^2 - 2x + 3 = 0 \).To find the zeros, apply the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a = 2 \), \( b = -2 \), and \( c = 3 \). Calculate the discriminant:\[b^2 - 4ac = (-2)^2 - 4(2)(3) = 4 - 24 = -20\]With a negative discriminant, the roots are complex. Substitute back into the quadratic formula to solve:\[ x = \frac{2 \pm \sqrt{-20}}{4} = \frac{2 \pm 2i\sqrt{5}}{4} = \frac{1 \pm i\sqrt{5}}{2}\]Thus, the polynomial has two complex zeros: \( \frac{1 + i\sqrt{5}}{2} \) and \( \frac{1 - i\sqrt{5}}{2} \). These complex roots show up as a conjugate pair due to the nature of the quadratic equation with real coefficients.