91Ó°ÊÓ

The Rational Root Theorem is a helpful tool when dealing with polynomials. It allows us to find possible rational roots by checking certain factors. Specifically, the theorem states: If a polynomial has a rational root \( \frac{p}{q} \), then \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
\[ x^3 - x - 6 \]
For the given polynomial, the constant term is 6 and the leading coefficient is 1. This means that any rational root of the polynomial must be a simple factor of 6.
Possible rational roots, therefore, include:

• \( \pm 1 \)
• \( \pm 2 \)
• \( \pm 3 \)
• \( \pm 6 \)

Once we list all potential rational roots, the next step is to test these values in the polynomial to determine which, if any, are actual roots. This involves substituting each potential root into the polynomial and checking if it results in zero.

Once we suspect a potential rational root, we use synthetic division to confirm it and reduce the polynomial for further analysis. Synthetic division is a simplified form of polynomial division, particularly when dividing by a linear factor. Let's say we found that 2 is indeed a root. We can proceed with synthetic division using the divisor \( x - 2 \).

Start with the coefficients of the polynomial, which in this case are \( 1, 0, -1, \) and \( -6 \). Begin by writing these coefficients in a row, and follow these steps:

• Bring down the leading coefficient (1).
• Multiply it by the root (2), and add the result under the next coefficient (0). This gives 2.
• Continue this process to yield the next coefficients, which manipulate the original polynomial by the suspected root.
• The final number in this division process should be zero if 2 is a true root.

This process not only verifies that 2 is a root but also simplifies the original polynomial, condensing it to \( x^2 + 2x + 3 \) for further analysis.

After using synthetic division, we're left with a simplified polynomial to continue our search for zeros, potentially including complex roots. For the expression \( x^2 + 2x + 3 = 0 \), we apply the quadratic formula to find its roots. The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Given the coefficients \( a = 1 \), \( b = 2 \), and \( c = 3 \), the first step is calculating the discriminant \( b^2 - 4ac \). In our case:
\[ 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8 \]
A negative discriminant value indicates the presence of complex roots. Integrating these into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{-8}}{2} = \frac{-2 \pm 2i \sqrt{2}}{2} \]
Simplifying yields the complex roots:
\[ x = -1 \pm i \sqrt{2} \]
This reveals that the polynomial not only has the real root 2 but also two complex roots.