To solve the problem of dividing the polynomial \(a^{3x} - 26a^{2x} + 156a^{x} - 216\) by \(a^{2x} - 24a^{x} + 108\), we will use polynomial long division. Let's break this down step-by-step. ### Step 1: Set Up the Division We are dividing: \[ \text{Dividend: } a^{3x} - 26a^{2x} + 156a^{x} - 216 \] by \[ \text{Divisor: } a^{2x} - 24a^{x} + 108. \] ### Step 2: Perform Polynomial Long Division 1. **Divide the leading term of the dividend by the leading term of the divisor**: \[ \frac{a^{3x}}{a^{2x}} = a^{x}. \] This means our first term in the quotient is \(a^{x}\). 2. **Multiply the entire divisor by \(a^{x}\)**: \[ a^{x}(a^{2x} - 24a^{x} + 108) = a^{3x} - 24a^{2x} + 108a^{x}. \] 3. **Subtract this result from the original dividend**: \[ (a^{3x} - 26a^{2x} + 156a^{x} - 216) - (a^{3x} - 24a^{2x} + 108a^{x}). \] This simplifies to: \[ -26a^{2x} + 24a^{2x} + 156a^{x} - 108a^{x} - 216 = -2a^{2x} + 48a^{x} - 216. \] 4. **Now, repeat the process with the new polynomial**: - Divide the leading term \(-2a^{2x}\) by \(a^{2x}\): \[ \frac{-2a^{2x}}{a^{2x}} = -2. \] So, the next term in the quotient is \(-2\). 5. **Multiply the entire divisor by \(-2\)**: \[ -2(a^{2x} - 24a^{x} + 108) = -2a^{2x} + 48a^{x} - 216. \] 6. **Subtract this from the current polynomial**: \[ (-2a^{2x} + 48a^{x} - 216) - (-2a^{2x} + 48a^{x} - 216) = 0. \] ### Step 3: Conclusion of Division Since the remainder is \(0\), we conclude that: \[ \frac{a^{3x} - 26a^{2x} + 156a^{x} - 216}{a^{2x} - 24a^{x} + 108} = a^{x} - 2. \] ### Final Answer Thus, the correct option is: **A. \(ax - 2\)**. ### Explanation of Other Options - **B. \(ax + 2\)**: This option suggests that the constant term is positive, which contradicts our result of \(-2\). - **C. \(ax - 8\)**: This option has an incorrect constant term, as we found the constant term to be \(-2\). - **D. \(ax - 6\)**: Similar to option C, this option also has an incorrect constant term. ### Revision Summary - Use polynomial long division to divide polynomials. - Divide the leading terms to find the first term of the quotient. - Multiply the divisor by the quotient term and subtract from the dividend. - Repeat until the remainder is zero or the degree of the remainder is less than the degree of the divisor. This methodical approach ensures clarity and accuracy in polynomial division.