Use synthetic substitution to find f(-3) for f(x) = 2x^3 - 6x^2 - 5x + 7.
_3|2 -6 -5 7
6 36 93
______________
2 0 31|100
then I used direct substitution and just plugged -3 in for x and the final answer is f(-3) = 86
To use synthetic substitution, follow these steps:
1. Write down the coefficients of the polynomial in descending order.
For f(x) = 2x^3 - 6x^2 - 5x + 7, the coefficients are 2, -6, -5, and 7.
2. Set up the synthetic division table.
- Place the constant term (7) on the top row of the table.
- Place the coefficient of the highest power of x (2) to the left of the constant term.
- Leave a space for the remaining coefficients.
The table should look like this:
| 2 -6 -5 7
3. Begin synthetic division.
- Bring down the first coefficient (2) below the line.
- Multiply it by the value you are substituting (-3 in this case).
- Write the result under the next coefficient (-6).
- Add the numbers diagonally and write the sum under the line.
The synthetic division should look like this:
|-3 3 -6 93
___________
2 -6 -5 7
4. Continue synthetic division.
- Repeat the steps.
- In the last step, the resulting value (100) is written above the line.
The completed synthetic division should look like this:
|-3 3 -6 93
___________
2 -6 -5 7
6 36 93
___________
2 0 31 100
5. Interpret the result.
- The numbers on the bottom row represent the coefficients of the resulting polynomial after substitution.
- In this case, the polynomial after substitution is 2x^2 + 31x + 100.
- The last value, 100, is the remainder.
Therefore, f(-3) = 100.
It seems there might be a mistake in your calculation for the final answer. The correct answer should be f(-3) = 100, not 86.
_-3/ 2 -6 -5 7
_________-6___36__-93__
2 -12 31 /-86
You left off the negative sign on your 3.