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+x²−4x−4

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Identify the polynomial function: \(f(x) = x^{3} + x^{2} - 4x - 4\).

List all possible rational zeros using the Rational Root Theorem. The possible rational zeros are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (-4) and \(q\) divides the leading coefficient (1). So, possible zeros are \(\pm1\), \(\pm2\), and \(\pm4\).

Use synthetic division to test each possible rational zero. Start by setting up synthetic division with one of the possible zeros (for example, \(x = 1\)) and divide the polynomial to see if the remainder is zero.

Once you find a zero that gives a remainder of zero, that value is an actual zero of the polynomial. The quotient from the synthetic division will be a quadratic polynomial.

Use the quadratic formula or factoring to find the remaining zeros from the quadratic quotient obtained in the previous step.

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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 factorization.

Factoring Polynomials and Finding Zeros

Once a zero is found, the polynomial can be factored using the quotient from synthetic division. This reduced polynomial is then solved to find remaining zeros, which may be real or complex, completing the factorization and root-finding process.

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+x2−4x−4