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^3−2x^2−11x+12

Verified step by step guidance

1. Step 1: Identify the possible rational zeros using the Rational Root Theorem. The possible rational zeros are the factors of the constant term (12) divided by the factors of the leading coefficient (1).
2. Step 2: List the factors of 12, which are ±1, ±2, ±3, ±4, ±6, ±12. Since the leading coefficient is 1, the possible rational zeros are ±1, ±2, ±3, ±4, ±6, ±12.
3. Step 3: Use synthetic division to test each possible rational zero. Start with x = 1 and perform synthetic division on the polynomial f(x) = x^3 - 2x^2 - 11x + 12.
4. Step 4: If the remainder is 0, then the tested value is a zero of the polynomial. If not, continue testing the next possible rational zero using synthetic division.
5. Step 5: Once an actual zero is found, use the quotient from the synthetic division to factor the polynomial further and find the remaining zeros by solving the resulting quadratic equation.

Key Concepts

Rational Root Theorem

The Rational Root Theorem provides a method for identifying all possible rational zeros of a polynomial function. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem helps narrow down the candidates for testing as potential zeros.

Synthetic Division

Synthetic division is a simplified form of polynomial long division that allows for efficient division of a polynomial by a linear factor. It involves using the coefficients of the polynomial and the potential zero to perform the division, yielding a quotient and a remainder. If the remainder is zero, the tested value is indeed a zero of the polynomial.

Finding Remaining Zeros

Once an actual zero is found using synthetic division, the polynomial can be expressed as a product of the linear factor corresponding to the zero and a reduced polynomial. The remaining zeros can then be found by factoring or using the quadratic formula on the resulting polynomial, allowing for a complete solution to the original polynomial equation.