Final Answer:
The value of h(3) is -729.
Step-by-step explanation:
We can utilize the Remainder Theorem and Synthetic Division to find h(3).
Remainder Theorem: When a polynomial p(x) is divided by x - a, the remainder is p(a).
Synthetic Division: This method simplifies the division process using a table format.
Here's how to find h(3) through synthetic division:
Step 1: Set up the table:
| | 3 | -16 | 20 | 7 | -13 |
|---|---|---|---|---|---|
| **3** | | | | | |
Step 2: Bring down the first coefficient:
| | 3 | -16 | 20 | 7 | -13 |
|---|---|---|---|---|---|
| **3** | 3 | | | | |
Step 3: Multiply the top term by the divisor (3) and write the result below the second term:
| | 3 | -16 | 20 | 7 | -13 |
|---|---|---|---|---|---|
| **3** | 3 | 9 | | | |
Step 4: Add the two numbers in the second column and write the sum below the line:
| | 3 | -16 | 20 | 7 | -13 |
|---|---|---|---|---|---|
| **3** | 3 | 9 | | | |
| | | -7 | | | |
Step 5: Repeat steps 3 and 4 for the remaining terms:
| | 3 | -16 | 20 | 7 | -13 |
|---|---|---|---|---|---|
| **3** | 3 | 9 | -7 | | |
| | | -7 | 27 | -49 | |
| | | | -49 | 52 | -45 |
| | | | | -45 | -104 |
Step 6: The remainder is the last number in the bottom row:
The bottom row shows that the remainder after dividing h(x) by x - 3 is -104. Since the remainder theorem states that the remainder of p(x) divided by x - a is equal to p(a), we can conclude that h(3) = -104.