Math Problem Statement

List all possible rational zeros given by the Rational Zeros Theorem of the given polynomial Q(x) = x^4 - 4x^3 - 5x + 8.

Solution

To list all possible rational zeros using the Rational Zeros Theorem for the polynomial Q(x)=x44x35x+8Q(x) = x^4 - 4x^3 - 5x + 8, we follow these steps:

Rational Zeros Theorem:

The possible rational zeros of a polynomial are of the form: pq\frac{p}{q} where:

  • pp is a factor of the constant term (here, 88),
  • qq is a factor of the leading coefficient (here, 11 since the coefficient of x4x^4 is 1).

Step 1: Identify the factors of the constant term (8).

The factors of 8 are: ±1,±2,±4,±8\pm 1, \pm 2, \pm 4, \pm 8

Step 2: Identify the factors of the leading coefficient (1).

The factors of 1 are: ±1\pm 1

Step 3: Form the possible rational zeros.

The possible rational zeros are: factors of 8factors of 1=±1,±2,±4,±8\frac{\text{factors of } 8}{\text{factors of } 1} = \pm 1, \pm 2, \pm 4, \pm 8

Thus, the set of possible rational zeros is: ±1,±2,±4,±8\pm 1, \pm 2, \pm 4, \pm 8

Conclusion:

The correct option from the choices is (b).

Would you like more details or have any questions?

Here are 5 related questions you might want to explore:

  1. How do you determine if a given rational zero is an actual root of the polynomial?
  2. What are the steps for synthetic division when testing for rational roots?
  3. How does the Rational Zeros Theorem relate to the Fundamental Theorem of Algebra?
  4. Can the Rational Zeros Theorem be applied to polynomials with non-integer coefficients?
  5. What other methods can be used to find the roots of a polynomial apart from the Rational Zeros Theorem?

Tip: After listing all possible rational zeros, use synthetic or long division to check which ones are actual zeros of the polynomial!

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Rational Zeros Theorem

Formulas

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Theorems

Rational Zeros Theorem

Suitable Grade Level

Grades 9-12