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. x³−2x²−11x+12=0

Verified step by step guidance

1. Identify the polynomial given: \(x^{3} - 2x^{2} - 11x + 12 = 0\).
2. a) To list all possible rational roots, use the Rational Root Theorem. The possible roots are all factors of the constant term (12) divided by all factors of the leading coefficient (1). So, list all factors of 12: \(\pm1, \pm2, \pm3, \pm4, \pm6, \pm12\).
3. b) Test these possible roots by substituting them into the polynomial or by using synthetic division to find which are actual roots. Once you find a root, perform synthetic division to divide the polynomial by \((x - \text{root})\) to get a quotient polynomial of degree 2.
4. c) Use the quotient polynomial from part (b), which will be a quadratic, and solve it using factoring, completing the square, or the quadratic formula to find the remaining roots.
5. Combine the root found in part (b) with the roots from the quadratic in part (c) to write the complete solution set for the equation.

Key Concepts

Rational Root Theorem

The Rational Root Theorem helps identify all possible rational roots of a polynomial equation by considering factors of the constant term and the leading coefficient. These possible roots are expressed as ±(factors of constant term)/(factors of leading coefficient), providing a finite list to test.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial of the form (x - c). After finding one root, dividing the polynomial by (x - root) simplifies the equation, reducing its degree and making it easier to find remaining roots.

Solving Polynomial Equations

Solving polynomial equations involves finding all roots (solutions) where the polynomial equals zero. After identifying rational roots and factoring, remaining roots can be found by solving the reduced polynomial, which may involve factoring, quadratic formula, or other methods.