Final answer:
To find the polynomial given that (x - 2) is a factor, you can perform synthetic division using 2 as the root. Once you complete the division, you'll have a quadratic equation as the quotient, which you can solve with the quadratic formula to find the remaining roots.
Step-by-step explanation:
To find the polynomial f(x) = x³ + 4x² - 4x - 16 given that (x - 2) is a factor, we can use polynomial division or synthetic division. Since (x - 2) is a factor, f(2) = 0. Let's perform the synthetic division with 2:
- Write down the coefficients: 1 (for x³), 4 (for x²), -4 (for x), and -16 (constant).
- Bring down the leading coefficient: 1.
- Multiply 1 by 2 (the root we are using for the division) and write the result under the next coefficient: 4 (the coefficient of x²).
- Add the numbers in the second column to get the new coefficient for x², which is (4 + 2 * 1) = 6.
- Repeat this process for the coefficients of x and the constant term.
The result of the synthetic division will provide the coefficients of the quotient polynomial. After performing synthetic division, we will have the quotient which is a quadratic equation, and we can use the quadratic formula to find the other roots if necessary.