To solve the problem, we need to find the coefficients \( a \), \( b \), and \( c \) in the polynomial \( ax^3 + bx^2 + cx - 1 \) given that \( (x - 1) \), \( (x + 1) \), and \( (x - 2) \) are factors of this polynomial. ### Step-by-Step Explanation 1. **Understanding Factors**: If \( (x - 1) \), \( (x + 1) \), and \( (x - 2) \) are factors of the polynomial, then the polynomial must equal zero when \( x = 1 \), \( x = -1 \), and \( x = 2 \). This means we can substitute these values into the polynomial to create a system of equations. 2. **Setting Up the Equations**: - For \( x = 1 \): \[ a(1)^3 + b(1)^2 + c(1) - 1 = 0 \implies a + b + c - 1 = 0 \implies a + b + c = 1 \quad \text{(Equation 1)} \] - For \( x = -1 \): \[ a(-1)^3 + b(-1)^2 + c(-1) - 1 = 0 \implies -a + b - c - 1 = 0 \implies -a + b - c = 1 \quad \text{(Equation 2)} \] - For \( x = 2 \): \[ a(2)^3 + b(2)^2 + c(2) - 1 = 0 \implies 8a + 4b + 2c - 1 = 0 \implies 8a + 4b + 2c = 1 \quad \text{(Equation 3)} \] 3. **Solving the System of Equations**: We now have a system of three equations: - \( a + b + c = 1 \) (Equation 1) - \( -a + b - c = 1 \) (Equation 2) - \( 8a + 4b + 2c = 1 \) (Equation 3) Let's solve these equations step by step. **From Equation 1**: \[ c = 1 - a - b \quad \text{(Substituting into other equations)} \] **Substituting into Equation 2**: \[ -a + b - (1 - a - b) = 1 \implies -a + b - 1 + a + b = 1 \implies 2b - 1 = 1 \implies 2b = 2 \implies b = 1 \] **Substituting \( b = 1 \) back into Equation 1**: \[ a + 1 + c = 1 \implies a + c = 0 \implies c = -a \] **Substituting \( b = 1 \) and \( c = -a \) into Equation 3**: \[ 8a + 4(1) + 2(-a) = 1 \implies 8a + 4 - 2a = 1 \implies 6a + 4 = 1 \implies 6a = -3 \implies a = -\frac{1}{2} \] **Finding \( c \)**: \[ c = -a = -\left(-\frac{1}{2}\right) = \frac{1}{2} \] 4. **Final Values**: We have found: - \( a = -\frac{1}{2} \) - \( b = 1 \) - \( c = \frac{1}{2} \) ### Conclusion Thus, the values of \( a \), \( b \), and \( c \) are: - \( a = -\frac{1}{2} \) - \( b = 1 \) - \( c = \frac{1}{2} \) The correct option is **A: -1/2, 1, 1/2**. ### Why Other Options Are Incorrect: - **Option B (1/2, 1, 1/2)**: This suggests \( a = 1/2 \), which contradicts our finding that \( a = -1/2 \). - **Option C (1/2, 1, -1/2)**: This incorrectly states \( c = -1/2 \) instead of \( c = 1/2 \). - **Option D (1/2, -1, 1/2)**: This incorrectly states \( b = -1 \) instead of \( b = 1 \). ### Revision Summary: - Factors of a polynomial lead to equations when substituted with their roots. - Solve the system of equations to find the coefficients. - Always check each option against the derived values to confirm correctness. - Be careful with signs and substitutions to avoid common pitfalls.