In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x^3+4x^2−3x−6

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1. Identify the polynomial function: $f(x)=x^3+4x^2-3x-6$.
2. List all possible rational zeros using the Rational Root Theorem: possible zeros are of the form ±(factors of constant term)/(factors of leading coefficient). Here, the constant term is 6 and the leading coefficient is 1, so possible rational zeros are ±1, ±2, ±3, ±6.
3. Use synthetic division to test each possible rational zero by dividing the polynomial by $x-c$ where $c$ is a candidate zero. Perform synthetic division step-by-step for each candidate until you find one that gives a remainder of zero, indicating an actual zero.
4. Once an actual zero is found, use the quotient polynomial from the synthetic division (which will be a quadratic) to find the remaining zeros by factoring or using the quadratic formula.
5. Write the complete factorization of the polynomial as $(x\; -\; c)(\; ext\{quotient\; polynomial\})$ and list all zeros found from these factors.

Key Concepts

Rational Root Theorem

The Rational Root Theorem helps identify all possible rational zeros of a polynomial 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 to test for actual zeros.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form (x - c). It efficiently tests whether a candidate root is an actual zero by checking if the remainder is zero. It also produces the quotient polynomial for further factorization.

Factoring Polynomials and Finding Zeros

Once a zero is found, the polynomial can be factored using the quotient from synthetic division. The reduced polynomial is then solved for remaining zeros, which may be real or complex. This step completes the factorization and identifies all roots of the polynomial.