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)=2x³−3x²−11x+6

Verified step by step guidance

1. Identify the polynomial function: \(f(x) = 2x^{3} - 3x^{2} - 11x + 6\).
2. List all possible rational zeros using the Rational Root Theorem. These are all fractions \(\frac{p}{q}\) where \(p\) divides the constant term (6) and \(q\) divides the leading coefficient (2). So, possible values of \(p\) are \(\pm1, \pm2, \pm3, \pm6\) and possible values of \(q\) are \(\pm1, \pm2\). Therefore, the possible rational zeros are \(\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}\).
3. Use synthetic division to test each possible rational zero by substituting them into the polynomial. Start with one candidate zero, perform synthetic division, and check if the remainder is zero. If the remainder is zero, that candidate is an actual zero of the polynomial.
4. Once an actual zero is found, use the quotient polynomial from the synthetic division (which will be of degree 2) to find the remaining zeros. This can be done by factoring the quadratic or using the quadratic formula if necessary.
5. Summarize the zeros found: the actual zero from synthetic division and the zeros from the quadratic factor, which together give all zeros of the original cubic polynomial.

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 for testing actual zeros.

Synthetic Division

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

Factoring Polynomials and Finding Zeros

Once a zero is found using synthetic division, the quotient polynomial can be factored further or solved using other methods (like quadratic formula) to find remaining zeros. Understanding how to factor or solve lower-degree polynomials is essential to completely determine all roots of the original polynomial.