Let's solve the polynomial function [tex]\( f(x) = x^3 - 4x^2 - 3x + 18 \)[/tex] at [tex]\( x = 10 \)[/tex] using synthetic division.
### Step-by-Step Solution:
Step 1: Set up synthetic division.
1. Write down the coefficients of the polynomial.
The polynomial [tex]\( f(x) = x^3 - 4x^2 - 3x + 18 \)[/tex] has coefficients [tex]\( [1, -4, -3, 18] \)[/tex].
2. Write the value of [tex]\( x \)[/tex] for which we are calculating the polynomial.
We need to calculate [tex]\( f(10) \)[/tex].
Step 2: Perform synthetic division.
1. Write down the coefficients.
The coefficients are [tex]\( 1, -4, -3, 18 \)[/tex].
2. Bring down the leading coefficient to the bottom row.
So, we bring down [tex]\( 1 \)[/tex].
```
10 | 1 -4 -3 18
| 10 60 570
----------------
1 6 57 588
```
3. Multiply the root [tex]\( 10 \)[/tex] by the value just written on the bottom row, and then write this product in the next column of the second row.
- Multiply [tex]\( 10 \times 1 = 10 \)[/tex] and write it under [tex]\( -4 \)[/tex].
- Add the column values [tex]\( -4 + 10 = 6 \)[/tex].
4. Repeat the process for each remaining coefficient.
- For the next column:
- Multiply [tex]\( 10 \times 6 = 60 \)[/tex] and write it under [tex]\( -3 \)[/tex].
- Add the column values [tex]\( -3 + 60 = 57 \)[/tex].
- For the last column:
- Multiply [tex]\( 10 \times 57 = 570 \)[/tex] and write it under [tex]\( 18 \)[/tex].
- Add the column values [tex]\( 18 + 570 = 588 \)[/tex].
Step 3: Interpret the result.
The bottom row now reads [tex]\( 1, 6, 57, 588 \)[/tex]. The last number [tex]\( 588 \)[/tex] is the value of the polynomial evaluated at [tex]\( x = 10 \)[/tex].
### Conclusion:
[tex]\( f(10) = 588 \)[/tex]
