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. x³−10x−12=0

Verified step by step guidance

1. Identify the polynomial equation given: $x{3}^{}-10x-12=0$.
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 and $q$ as a factor of the leading coefficient. Here, the constant term is -12 and the leading coefficient is 1.
3. List the factors of the constant term (-12): ±1, ±2, ±3, ±4, ±6, ±12. Since the leading coefficient is 1, its factors are ±1. Therefore, the possible rational roots are simply the factors of -12: ±1, ±2, ±3, ±4, ±6, ±12.
4. For part (b), test these possible roots by substituting them into the polynomial or by using synthetic division to find which one is an actual root. The root that yields a zero remainder is a root of the polynomial.
5. For part (c), once you find a root from part (b), use synthetic division or polynomial division to divide the original polynomial by $(x\; -\; r)$, where $r$ is the root found. This will give you a quadratic quotient. Then, solve the quadratic equation 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. For a polynomial with integer coefficients, any rational root is a fraction p/q, where p divides the constant term and q divides the leading coefficient.

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 or dividing the polynomial, solving the resulting lower-degree polynomial or quadratic equation yields the remaining roots. Techniques include factoring, using the quadratic formula, or further root testing, allowing complete solution of the original equation.