In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x^3−10x−12=0

Verified step by step guidance

1. Step 1: Identify the polynomial equation given: \(x^3 - 10x - 12 = 0\).
2. Step 2: Use the Rational Root Theorem to list all possible rational roots. The possible rational roots are the factors of the constant term (-12) divided by the factors of the leading coefficient (1).
3. Step 3: List the factors of -12: \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\). Since the leading coefficient is 1, the possible rational roots are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).
4. Step 4: Use synthetic division or polynomial division to test these possible rational roots and find one actual root of the equation.
5. Step 5: Once a root is found, use the quotient from the division to factor the polynomial further and find the remaining roots.

Key Concepts

Rational Root Theorem

The Rational Root Theorem states that any rational solution of a polynomial equation, in the form of p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem helps in identifying all possible rational roots of a polynomial, which can then be tested to find actual roots.

Synthetic Division

Synthetic division is a simplified form of polynomial long division that allows for the division of a polynomial by a linear factor. It is particularly useful for finding the quotient and remainder when a polynomial is divided by a binomial of the form (x - r), where r is a root. This method streamlines the process of finding remaining roots after identifying one root.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, which can be linear or quadratic expressions. Once a polynomial is factored, it becomes easier to solve for its roots by setting each factor equal to zero. This concept is essential for simplifying polynomial equations and finding all possible solutions.