Final answer:
To find the quotient and remainder of the expression (7ⁿ⁴ + 59ⁿ³ - 41nⁿ² - 51n - 48) ÷ (n + 9), perform long division step by step, starting with the highest power term. Keep dividing and subtracting until you reach the lowest power term, which will be the remainder.
Step-by-step explanation:
To find the quotient and remainder of the expression (7ⁿ⁴ + 59ⁿ³ - 41nⁿ² - 51n - 48) ÷ (n + 9), we need to perform long division. Let's go step by step:
- Start by dividing the highest power term, which is 7ⁿ⁴, by n+9. The quotient is 7ⁿ³.
- Multiply the quotient (7ⁿ³) by (n + 9), which gives us 7ⁿ⁴ + 63ⁿ³.
- Subtract this result (7ⁿ⁴ + 63ⁿ³) from the original expression (7ⁿ⁴ + 59ⁿ³ - 41nⁿ² - 51n - 48). We get: (59ⁿ³ - 63ⁿ³ - 41nⁿ² - 51n - 48).
- Repeat the process with the next highest power term, which is 59ⁿ³. Divide it by n+9 to get the quotient.
- Multiply the new quotient by (n + 9) and subtract it from (59ⁿ³ - 63ⁿ³ - 41nⁿ² - 51n - 48).
- Continue this process with each term until you have no more powers of n. The last term, after you divide -48 by n+9, will be the remainder.
Remember to distribute the minus sign when subtracting expressions, and simplify the results as much as possible. Good luck!