Introduction

Understanding the process of identifying rational zeros and factoring polynomials is fundamental to solving polynomial equations. In this article, we will explore how to determine the possible rational zeros and then find all the zeros of the polynomial equation x^3 - 5x^2 - x - 5. We will use the Rational Root Theorem and demonstrate the steps involved in the process, including factorization and verification through graphing and an example with R software.

Step-by-Step Process

1. Identifying Possible Rational Zeros

The Rational Root Theorem is a powerful tool for finding the possible rational zeros of a polynomial. According to this theorem, the potential rational zeros of a polynomial ax^n bx^(n-1) ... k are given by the ratio p/q, where p is a factor of the constant term k, and q is a factor of the leading coefficient a.

Step 1: Identifying the Factors

In our polynomial f(x) x^3 - 5x^2 - x - 5:

The factors of -5 are pm 1, pm 5. For the leading coefficient, the factors are pm 1.

Step 2: Possible Rational Zeros

Based on the Rational Root Theorem, the possible rational zeros are the ratios of the factors of the constant term and the leading coefficient. In this case, the possible rational zeros are:

2. Testing the Possible Rational Zeros

Now we will test these possible zeros by substituting them back into the polynomial to see if they satisfy the equation.

Testing x 1

Let's substitute x 1 into the polynomial:

f(1)  1^3 - 5(1)^2 - 1 - 5  1 - 5 - 1 - 5  0

Since f(1) 0, x 1 is a zero.

Testing x -1

Let's substitute x -1 into the polynomial:

f(-1)  (-1)^3 - 5(-1)^2 - (-1) - 5  -1 - 5   1 - 5  0

Since f(-1) 0, x -1 is a zero.

Testing x 5

Let's substitute x 5 into the polynomial:

f(5)  5^3 - 5(5)^2 - 5 - 5  125 - 125 - 5 - 5  0

Since f(5) ≠ 0, x 5 is not a zero.

Testing x -5

Let's substitute x -5 into the polynomial:

f(-5)  (-5)^3 - 5(-5)^2 - (-5) - 5  -125 - 125   5 - 5  0

Since f(-5) 0, x -5 is a zero.

3. Finding All the Zeros

We have found three rational zeros: x 1, -1, -5. Using these zeros, we can factor the polynomial:

f(x)  (x - 1)(x   1)(x   5)

Let's confirm the factorization through multiplication:

Multiplying the Factors

First, multiply (x - 1)(x 1):

(x - 1)(x   1)  x^2 - 1

Now, multiply by (x 5):

(x^2 - 1)(x   5)  x^3   5x^2 - x - 5

This confirms that our factorization is correct.

4. Conclusion

The polynomial x^3 - 5x^2 - x - 5 has the following zeros:

As always, it is recommended to graph the function to get a visual estimate of the real roots. From the graph, it is clear that the rational roots are near x -3, -1, and 3.

Verification Using R Software

The polyroot function from R software can also be used to verify the zeros of the polynomial. The results should match the zeros found through the Rational Root Theorem and factorization.

Conclusion

In summary, understanding and applying the Rational Root Theorem is a crucial step in solving polynomial equations. By systematically testing the possible rational zeros and confirming the factorization, we can accurately determine the roots of polynomials. This process not only deepens our mathematical understanding but also provides practical tools for problem-solving in various fields.