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 given: \(x^{3} - 10x - 12 = 0\). Notice that the polynomial is cubic, and the constant term is \(-12\), while the leading coefficient is \$1$.
2. Use the Rational Root Theorem to list all possible rational roots. According to the theorem, possible rational roots are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term and \(q\) divides the leading coefficient. Since the leading coefficient is 1, possible roots are the divisors of \(-12\): \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).
3. Test each possible rational root by substituting into the polynomial to find which one(s) yield zero. This can be done by direct substitution or synthetic division.
4. Once a root is found, use synthetic division (or polynomial long division) to divide the original polynomial by \((x - r)\), where \(r\) is the root found. This will give a quadratic quotient.
5. Solve the quadratic quotient using factoring, completing the square, or the quadratic formula to find the remaining roots of 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). 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 applying methods like the quadratic formula. This step finds all roots, including irrational or complex ones, completing the solution to the equation.