Math Problem Statement

Use the rational zeros theorem to list the potential rational zeros of the polynomial function. Do not attempt to find the zeros. f(x) = 2x^7 - x^5 + 5x^4 + 28 Find the potential rational zeros. Choose the correct answer below: A. -1, 1, -2, 2, -4, 4, -7, 7, -14, 14, -28, 28, -1/2, 1/2, -1/28, 1/28 B. -1, 1, -2, 2, -4, 4, -7, 7, -14, 14, -28, 28, -1/2, 1/2, -7/2, 7/2 C. -1, 1, -2, 2, -4, 4, -7, 7, -28, 28, -1/2, 1/2, -1/28, 1/28 D. -1, 1, -2, 2, -4, 4, -7, 7, -28, 28, -1/2, 1/2, -1/28, 1/28

Solution

To solve this problem, we need to apply the Rational Zeros Theorem to the polynomial function:

f(x)=2x7x5+5x4+28f(x) = 2x^7 - x^5 + 5x^4 + 28

Step-by-Step Explanation

  1. Identify the coefficients:

    • The leading coefficient is 22 (coefficient of x7x^7).
    • The constant term is 2828.
  2. Rational Zeros Theorem:

    • The potential rational zeros are in the form pq\frac{p}{q}, where:
      • pp is a factor of the constant term (28).
      • qq is a factor of the leading coefficient (2).
  3. Factors of the constant term (28): ±1,±2,±4,±7,±14,±28\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28

  4. Factors of the leading coefficient (2): ±1,±2\pm 1, \pm 2

  5. Form all combinations pq\frac{p}{q}: ±1,±2,±4,±7,±14,±28,±12,±72,±142,±282\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{2}, \pm \frac{7}{2}, \pm \frac{14}{2}, \pm \frac{28}{2} Simplifying the fractions: ±1,±2,±4,±7,±14,±28,±12,±72\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{2}, \pm \frac{7}{2}

Answer Verification

From the options provided:

  • Option D matches the correct potential rational zeros list, including ±128\pm \frac{1}{28}.

Correct Answer: D

Would you like a detailed explanation on any part of the solution?

Related Questions:

  1. How do you find factors of a number for the Rational Zeros Theorem?
  2. What happens if the polynomial has a leading coefficient of 1?
  3. Can the Rational Zeros Theorem identify all real zeros of a polynomial?
  4. What is the next step if you want to verify whether a potential zero is an actual zero?
  5. How do you perform synthetic division to test potential zeros?

Tip:

Always check the simplification of fractions when listing potential rational zeros to avoid redundancy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Rational Zeros Theorem

Formulas

Rational Zeros Theorem: p/q, where p is a factor of the constant term and q is a factor of the leading coefficient

Theorems

Rational Zeros Theorem

Suitable Grade Level

Grade 11-12