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The Rational Root Theorem is a helpful tool when solving polynomial equations. It helps you find the possible rational roots of a polynomial. These roots are potential candidates, which you can test to see if they are actual roots of the equation. The theorem states that any possible rational root \( \frac{p}{q} \) of a polynomial equation must be a fraction, where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

In the given problem, the polynomial is \( x^{4}-15x^{2}-20x-6 \).
To find the possible rational roots:

• Identify the constant term: \( -6 \)
• Identify the leading coefficient: \( 1 \)
• List factors of \( -6 \): \( \text{±1, ±2, ±3, ±6} \)
• The factors of \( 1 \) are: \( \text{±1} \)

Therefore, the possible rational roots are \( \text{±1, ±2, ±3, ±6} \). By using the Rational Root Theorem, we narrow down the possible candidates for true roots.

Synthetic Division is a simplified method to perform polynomial division, particularly useful when dividing by linear factors of the form \(x - c\). It is faster and easier than long division.

For the polynomial \( x^{4}-15x^{2}-20x-6 \) given in the exercise, to verify if a possible root like \( \text{-3} \) is an actual root, we use synthetic division.
Follow these steps to use synthetic division:

• Write down the coefficients: \( 1, 0, -15, -20, -6 \)
• Set up the synthetic division with \( -3 \) as the divisor:

Here's the step-by-step:

• Bring down the first coefficient (1).
• Multiply the divisor (-3) by the first number obtained (1) and write the result under the next coefficient.
• Add the column to get a new bottom number.
• Repeat the process until you reach the final coefficient.

When \( -3 \) is used, the remainder ends up being zero, confirming that \( x + 3 \) is a factor. The quotient here would be \( x^3 - 3x^2 - 6x - 2 \).

After simplifying the polynomial to a quadratic equation, you may need to solve it using the Quadratic Formula. This formula is an easy way to find roots for any quadratic equation of the form \( ax^2 + bx + c = 0 \).

The Quadratic Formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In the given problem, we eventually arrive at the quadratic equation: \( x^2 - 4x - 2 = 0 \).

Using the coefficients from this equation, apply the Quadratic Formula:

• \( a = 1 \)
• \( b = -4 \)
• \( c = -2 \)

Now plug these into the formula:

• \