A. List all possible rational roots. b. Use synthetic...

a. List all possible rational roots.
b. Use synthetic division to test the possible rational roots and find an actual root.
c. Use the quotient from part (b) to find the remaining roots and solve the equation.

$6 x^{3}+25 x^{2}-24 x+5=0$

Key Concepts

Polynomial Factorization and Solving Equations

Once a root is found, the polynomial can be factored into a product of lower degree polynomials. This factorization simplifies the equation and allows for the remaining roots to be determined, often by applying methods such as the quadratic formula for a resulting quadratic factor.

Factor Theorem

The Factor Theorem states that a polynomial has a factor (x - r) if and only if the polynomial evaluates to zero at r. This is a critical concept in polynomial factorization, as it connects the existence of roots to the factors of the polynomial.

Rational Root Theorem

The Rational Root Theorem provides a method for listing all possible rational roots of a polynomial equation with integer coefficients. It does so by stating that any possible rational root, expressed in lowest terms as p/q, must have a numerator p that is a factor of the constant term and a denominator q that is a factor of the leading coefficient.

Synthetic Division

Synthetic division is a streamlined method of dividing a polynomial by a binomial of the form (x - r). It is particularly useful for testing whether a candidate root is an actual root of the polynomial by quickly determining the remainder, which if zero, confirms that the candidate is indeed a root.

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