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

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Identify the polynomial function: \(f(x) = x^{3} - 2x^{2} - 11x + 12\).

List all possible rational zeros using the Rational Root Theorem. These are all factors of the constant term (12) divided by factors of the leading coefficient (1). So, possible rational zeros are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).

Use synthetic division to test each possible rational zero by substituting them into the polynomial. Start with one candidate, perform synthetic division, and check if the remainder is zero. If the remainder is zero, that candidate is an actual zero of the polynomial.

Once an actual zero is found, the quotient from the synthetic division will be a quadratic polynomial. Write down this quotient polynomial.

Solve the quadratic polynomial from the quotient using factoring, completing the square, or the quadratic formula to find the remaining zeros of the original polynomial.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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), providing a finite list to test.

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, and it produces a quotient polynomial for further analysis.

Factoring Polynomials and Finding Zeros

Once a root is found using synthetic division, the quotient polynomial can be factored further or solved using other methods to find remaining zeros. This step breaks down the polynomial into simpler factors, allowing all roots (real or complex) to be identified.

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