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^3−2x^2−11x+12=0

Verified step by step guidance

1. Step 1: Identify the polynomial and its coefficients. The given polynomial is \(x^3 - 2x^2 - 11x + 12 = 0\). The coefficients are 1, -2, -11, and 12.
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, which are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\). 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 to test each possible rational root from Step 3 to find one that results in a remainder of 0. This will confirm it as a root of the polynomial.
5. Step 5: Once a root is found, use the quotient from synthetic division to factor the polynomial further and find the remaining roots. Solve the resulting quadratic equation, if necessary, to find all roots of the original polynomial.

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 process helps in reducing the polynomial degree and finding remaining roots.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler polynomial factors. This is essential for solving polynomial equations, as it allows us to set each factor equal to zero to find the roots. Techniques include grouping, using the quadratic formula, or applying special product formulas.