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)=x³−2x²−11x+12

Verified step by step guidance

1. Identify the polynomial function: $f(x)=x^3-2x^2-11x+12$.
2. List all possible rational zeros using the Rational Root Theorem: possible zeros are all factors of the constant term (12) divided by factors of the leading coefficient (1). So, possible rational zeros are ±1, ±2, ±3, ±4, ±6, ±12.
3. Use synthetic division to test each possible rational zero by dividing the polynomial by $x-$ where is each candidate zero. Perform synthetic division until you find a zero that gives a remainder of 0.
4. Once an actual zero is found, write the quotient polynomial from the synthetic division. This quotient will be a quadratic polynomial since the original was cubic.
5. Solve the quadratic quotient polynomial using factoring, completing the square, or the quadratic formula to find the remaining zeros of the original 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. For a polynomial f(x), possible rational roots are ±(factors of constant term) / (factors of leading coefficient). This 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. If the remainder is zero, the divisor corresponds to a factor of the polynomial.

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. This step breaks down the polynomial into simpler factors, revealing all roots of the function.