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. For part (a), use the Rational Root Theorem to list all possible rational roots. The theorem states that any rational root, expressed as $\backslash frac\{p\}\{q\}$, must have $p$ as a factor of the constant term (12) and $q$ as a factor of the leading coefficient (1).
3. List the factors of the constant term 12: ±1, ±2, ±3, ±4, ±6, ±12. Since the leading coefficient is 1, the possible rational roots are simply these factors: ±1, ±2, ±3, ±4, ±6, ±12.
4. For part (b), use synthetic division or polynomial division to test these possible roots and find one that yields a zero remainder. This root is an actual root of the polynomial.
5. For part (c), use the quotient polynomial obtained from the division in part (b) (which will be quadratic) and solve it using factoring, completing the square, or the quadratic formula to find the remaining roots.

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). This theorem narrows down candidates for testing actual roots.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial of the form (x - r), where r is a root. This process simplifies the polynomial to a lower degree, making it easier to find remaining roots. Synthetic division is a streamlined method often used for this purpose.

Solving Polynomial Equations

After factoring the polynomial using known roots, the remaining polynomial can be solved by factoring further or using methods like the quadratic formula. Finding all roots involves combining rational roots found and solving the reduced polynomial to get all solutions to the equation.