To solve the equation \( x^3 - 5x^2 - x + 5 = 0 \), we will use a combination of techniques including factoring and the Rational Root Theorem. Let's go through the solution step-by-step. ### Step 1: Identify the Polynomial The equation we need to solve is: \[ x^3 - 5x^2 - x + 5 = 0 \] This is a cubic polynomial, and we are looking for its roots. ### Step 2: Use the Rational Root Theorem The Rational Root Theorem suggests that any rational solution (root) of the polynomial equation, in the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (5) and \( q \) is a factor of the leading coefficient (1). The factors of 5 are \( \pm 1, \pm 5 \). Since the leading coefficient is 1, the possible rational roots are: \[ \pm 1, \pm 5 \] ### Step 3: Test Possible Roots We will test these possible roots by substituting them into the polynomial. 1. **Testing \( x = 1 \)**: \[ 1^3 - 5(1^2) - 1 + 5 = 1 - 5 - 1 + 5 = 0 \] So, \( x = 1 \) is a root. 2. **Testing \( x = -1 \)**: \[ (-1)^3 - 5(-1)^2 - (-1) + 5 = -1 - 5 + 1 + 5 = 0 \] So, \( x = -1 \) is also a root. 3. **Testing \( x = 5 \)**: \[ 5^3 - 5(5^2) - 5 + 5 = 125 - 125 - 5 + 5 = 0 \] So, \( x = 5 \) is also a root. ### Step 4: Factor the Polynomial Since we have found three roots \( x = 1, -1, 5 \), we can express the polynomial as: \[ (x - 1)(x + 1)(x - 5) = 0 \] ### Step 5: Verify the Factorization To ensure our factorization is correct, we can expand: \[ (x - 1)(x + 1) = x^2 - 1 \] Now multiply by \( (x - 5) \): \[ (x^2 - 1)(x - 5) = x^3 - 5x^2 - x + 5 \] This confirms our factorization is correct. ### Step 6: State the Roots The roots of the equation \( x^3 - 5x^2 - x + 5 = 0 \) are: \[ x = 1, -1, 5 \] ### Conclusion: Correct Option The correct option is **A: 1, -1, or 5**. ### Explanation of Other Options - **Option B: 1, 1, or -5**: This is incorrect because it suggests that 1 is a double root and includes -5, which is not a root. - **Option C: -1, 1, or -5**: This is incorrect because it includes -5, which is not a root. - **Option D: 1, 1, or 5**: This is incorrect because it suggests that 1 is a double root, which is not the case. ### Revision Summary - The roots of the polynomial \( x^3 - 5x^2 - x + 5 = 0 \) are found using the Rational Root Theorem. - The roots are \( x = 1, -1, 5 \). - The polynomial can be factored as \( (x - 1)(x + 1)(x - 5) = 0 \). - The correct answer is option A: 1, -1, or 5.