ide. If the polynomial does not divide evenly, include the remainder as a fraction. \( \left(r^{4}-32 r^{2}-119 r\right) \div(r-7) \) a

To divide the polynomial \( r^{4}-32 r^{2}-119 r \) by \( r-7 \), you can use synthetic division or long division. When performing the division, you start by dividing the leading term of the dividend by the leading term of the divisor, which gives you \( r^3 \). After multiplying back and subtracting, you'll repeat this process with the resulting polynomial until you reach a degree lower than the divisor. After completing the division, you will find a quotient of \( r^3 + 7r^2 - 18r + 35 \) with a remainder of \( 106 \). Thus, the final answer is \( r^{3} + 7r^{2} - 18r + 35 + \frac{106}{r-7} \). Now you have not just the quotient but also the remainder, which provides a complete picture of the division! It's a fun way to see how polynomial division can reveal important information about the relationship between two mathematical expressions.