Question

Use synthetic division and the Remainder Theorem to find the indicated function value.
f left parenthesis x right parenthesis equals 4 x cubed minus 11 x squared plus 5 x minus 6​; f left parenthesis 1 right parenthesis
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Part 1
f left parenthesis 1 right parenthesis equals nothing
  

Answer 1

To find \( f(1) \) using synthetic division and the Remainder Theorem, we first set up synthetic division with \( f(x) = 4x^3 - 11x^2 + 5x - 6 \) and \( x = 1 \).

**Step 1: Setting up synthetic division**

1. Write down the coefficients of the polynomial:
- Coefficients: \( 4, -11, 5, -6 \)

2. Set up the synthetic division with \( 1 \) (the value for \( x \)):

```
1 | 4 -11 5 -6
|
```

**Step 2: Performing synthetic division**

1. Bring down the first coefficient:
```
1 | 4 -11 5 -6
|
| 4
```

2. Multiply \( 4 \) by \( 1 \) and write the result under the next coefficient:
```
1 | 4 -11 5 -6
| 4
| 4 -7
```

3. Add \( -11 \) and \( 4 \):
```
1 | 4 -11 5 -6
| 4 -7
| 4 -7 -2
```

4. Multiply \( -7 \) by \( 1 \) and write the result under the next coefficient:
```
1 | 4 -11 5 -6
| 4 -7
| 4 -7 -2
```

5. Add \( 5 \) and \( -7 \):
```
1 | 4 -11 5 -6
| 4 -7
| 4 -7 -2
```

6. Multiply \( -2 \) by \( 1 \) and write the result under the next coefficient:
```
1 | 4 -11 5 -6
| 4 -7 -2
| 4 -7 -8
```

7. Finally, add \( -6 \) and \( -2 \):
```
1 | 4 -11 5 -6
| 4 -7 -2
| 4 -7 -8
```

The last value, \( -8 \), is the remainder and this represents \( f(1) \).

**Final Answer:**
\[
f(1) = -8
\]