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. 6x³+25x²−24x+5=0

Verified step by step guidance

1. Identify the polynomial equation: $\mathrm{6x^3}+25\mathrm{x^2}-24x+5=0$.
2. For part (a), list all possible rational roots using the Rational Root Theorem. The possible roots are of the form $\backslash pm\; \backslash frac\{p\}\{q\}$, where $p$ divides the constant term (5) and $q$ divides the leading coefficient (6).
3. Determine the divisors of the constant term: $\backslash pm\; 1,\; \backslash pm\; 5$, and the divisors of the leading coefficient: $1,\; 2,\; 3,\; 6$. Then form all possible fractions $\backslash pm\; \backslash frac\{p\}\{q\}$ in simplest form.
4. For part (b), use synthetic division or polynomial division to test each possible rational root from part (a) until you 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) and solve it (either by factoring, quadratic formula, or other methods) to find the remaining roots of the original cubic 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. 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 for this purpose when dividing by linear factors.

Solving Polynomial Equations

After factoring or dividing the polynomial, solving the resulting lower-degree polynomial or quadratic involves finding roots using factoring, quadratic formula, or other algebraic methods. These roots, combined with previously found roots, provide the complete solution set to the original equation.