Certainly! Let's use synthetic division and the Remainder Theorem to find the value of [tex]\( f(-4) \)[/tex] for the polynomial [tex]\( f(x) = 3x^3 - 5x^2 - 8x + 5 \)[/tex].
Here are the steps:
1. Set up synthetic division: Write down the coefficients of the polynomial and the value you want to evaluate—in this case, [tex]\(-4\)[/tex].
Coefficients: [tex]\(3, -5, -8, 5\)[/tex]
Value to evaluate: [tex]\(-4\)[/tex]
2. Write the coefficients and the value you're evaluating to the left:
```
-4 | 3 | -5 | -8 | 5
```
3. Bring down the [tex]\(3\)[/tex] to start:
```
-4 | 3 | -5 | -8 | 5
└── 3
```
4. Multiply and add: Multiply the value you are evaluating (-4) by the number you just brought down (3), then add this result to the next coefficient (-5):
- Multiply: [tex]\(-4 \times 3 = -12\)[/tex]
- Add: [tex]\(-5 + (-12) = -17\)[/tex]
```
-4 | 3 | -5 | -8 | 5
└── 3 -17
```
5. Repeat the process: Multiply the value you are evaluating (-4) by the result you just obtained (-17) and then add to the next coefficient (-8):
- Multiply: [tex]\(-4 \times (-17) = 68\)[/tex]
- Add: [tex]\(-8 + 68 = 60\)[/tex]
```
-4 | 3 | -5 | -8 | 5
└── 3 -17 60
```
6. One last time: Multiply the value you are evaluating (-4) by the result you just obtained (60) and then add to the final coefficient (5):
- Multiply: [tex]\(-4 \times 60 = -240\)[/tex]
- Add: [tex]\(5 + (-240) = -235\)[/tex]
```
-4 | 3 | -5 | -8 | 5
└── 3 -17 60 -235
```
7. Interpret the result:
The last value on the row is the value of the polynomial at [tex]\( x = -4 \)[/tex]. Therefore, [tex]\( f(-4) = -235 \)[/tex].
So by using synthetic division and the Remainder Theorem, we find that [tex]\( f(-4) = -235 \)[/tex].
