Final answer:
The question involves using synthetic division to divide the polynomial x^3 + 21x^2 + 15 by x + 1. The outcome of the process is a quadratic polynomial x^2 + 20x - 20 with a remainder of -5.
Step-by-step explanation:
The question is asking to divide a polynomial, x3 + 21x2 + 15, by another polynomial, x + 1, using synthetic division. Synthetic division is a simplified form of polynomial division, used when dividing by a linear factor, and particularly useful when the divisor is in the form x - c.
To use synthetic division, we first identify the value that x needs to be for the divisor to equal zero, which is -1 in this case since x + 1 = 0 when x = -1. Next, we list the coefficients of the dividend: 1 (for x3), 21 (for x2), 0 (since there is no x term, we use a placeholder), and 15 (the constant term).
We then bring down the leading coefficient (1) and proceed with the synthetic division process:
- Multiply the value brought down by -1 and write the result under the next coefficient (21).
- Add this result to the next coefficient and write the sum underneath.
- Repeat the above two steps for the remaining coefficients.
Carrying out the steps, the synthetic division would look something like this:
-1 | 1 21 0 15
|___________________
| 1 20 -20 -5
The result of the division is x2 + 20x - 20 with a remainder of -5.