1. **Problem:** Divide the polynomial $x^3 - 7x - 17$ by $x - 8$. 2. **Formula and Rules:** Polynomial division is similar to long division with numbers. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by this result, subtract, and repeat. 3. **Step-by-step division:** - Divide $x^3$ by $x$ to get $x^2$. - Multiply $x^2$ by $(x - 8)$ to get $x^3 - 8x^2$. - Subtract: $(x^3 - 7x - 17) - (x^3 - 8x^2) = \cancel{x^3} - \cancel{x^3} + 8x^2 - 7x - 17 = 8x^2 - 7x - 17$. - Divide $8x^2$ by $x$ to get $8x$. - Multiply $8x$ by $(x - 8)$ to get $8x^2 - 64x$. - Subtract: $(8x^2 - 7x - 17) - (8x^2 - 64x) = \cancel{8x^2} - \cancel{8x^2} + ( -7x + 64x) - 17 = 57x - 17$. - Divide $57x$ by $x$ to get $57$. - Multiply $57$ by $(x - 8)$ to get $57x - 456$. - Subtract: $(57x - 17) - (57x - 456) = \cancel{57x} - \cancel{57x} + (-17 + 456) = 439$. 4. **Result:** Quotient is $x^2 + 8x + 57$ and remainder is $439$. 5. **Final answer:** $$\frac{x^3 - 7x - 17}{x - 8} = x^2 + 8x + 57 + \frac{439}{x - 8}$$