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The Rational Root Theorem is a useful tool for finding possible rational roots of polynomial equations. It states that any potential rational root of a polynomial is a fraction where the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient.

For our polynomial, \(f(x) = x^4 - x^3 - 19x^2 + 49x - 30\), the constant term is -30, and the leading coefficient is 1. Therefore, the list of possible rational roots includes all factors of -30, with no other numbers since the leading coefficient is 1.

This gives us possible roots: \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30\).

To find the actual roots, we test these values in the polynomial. This helps us narrow down our search for the exact roots of the equation.

Synthetic division is a simplified form of polynomial division that is specifically used to divide by a linear factor of the form \(x - c\). It’s faster and organized compared to traditional long division.

Let's go through synthetic division with our polynomial \(f(x)\) and a known root. For example, we found that \(x = 1\) is a root, so we divide \(x^4 - x^3 - 19x^2 + 49x - 30\) by \(x - 1\).

First, write down the coefficients of the polynomial: \(1, -1, -19, 49, -30\).

Next, follow these steps:

• Bring down the leading coefficient (1).
• Multiply it by the root (1), and add the result to the next coefficient: \(-1 + 1 = 0\).
• Repeat the multiplication and addition process until you complete the row.

The resulting coefficients from synthetic division give us the quotient polynomial: \(x^3 - 19x + 30\). This step-by-step process can be repeated for further factorization.

After using synthetic division, we may get a quadratic polynomial that we need to factor further.

For the polynomial \(x^2 + 2x - 15\), we look for two numbers that multiply to -15 (the constant term) and add to 2 (the coefficient of the linear term). These numbers are 5 and -3.

Thus, we can write \(x^2 + 2x - 15\) as \((x + 5)(x - 3)\).

This method allows us to break down complex polynomials into simpler factors that can be easily solved.

Combining all the factors from our step-by-step solution, we have \(f(x) = (x - 1)(x - 2)(x + 5)(x - 3)\). This complete factorization shows us all the roots of the original polynomial when set to zero.