Final answer:
Using synthetic division, the polynomial f(x) = x⁴ + 3x³ + 9x² - 2x - 2 is divided by x - 7. The sequence of operations involves multiplication, and addition to find the remainder, which is f(7). This process simplifies algebra compared to traditional long division.
Step-by-step explanation:
The question revolves around using synthetic division to divide the polynomial f(x) = x⁴ + 3x³ + 9x² - 2x - 2 by x - 7 (since we are evaluating for f(7)). Synthetic division is a simplified method of dividing a polynomial by a binomial of the form x - c. Here's the step-by-step process:
- Write down the coefficients of f(x): 1, 3, 9, -2, and -2.
- Place the number 7 (the zero of x - 7) to the left of a vertical bar and the coefficients to its right.
- Bring down the leading coefficient (1) to the bottom row.
- Multiply 7 by the number at the bottom row and write the result under the next coefficient.
- Add the numbers in the second column, and continue this process of multiplying by 7 and adding until you reach the end.
- The last number obtained in the bottom row will be the remainder f(7).
To check if the answer is reasonable, one might consider the values of the coefficients and the divisor (7) for potential computation errors.
The process simplifies algebra by avoiding the more extensive long division method and quickly finds the remainder, which in turn provides f(7).