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The problem asks to perform synthetic division of the polynomial \( 3x^3 - 17x^2 + 13x + 23 \) by \( x - 4 \), and express the result in the form:

\[
q(x) + \frac{r(x)}{b(x)}
\]
where \( q(x) \) is the quotient, \( r(x) \) is the remainder, and \( b(x) \) is the divisor.

### Step 1: Set up the synthetic division.

In synthetic division, we work with the coefficients of the polynomial \( 3x^3 - 17x^2 + 13x + 23 \) and the divisor \( x - 4 \).

The coefficients of the polynomial are: \( 3, -17, 13, 23 \).  
The root of the divisor \( x - 4 \) is \( 4 \).

We will now set up the synthetic division table:

```
4 |  3    -17    13    23
   |        (   )  (   )  (   )
   --------------------------
```

### Step 2: Perform synthetic division.

1. Bring down the first coefficient \( 3 \).

```
4 |  3    -17    13    23
   |        
   --------------------------
       3
```

2. Multiply \( 3 \) by \( 4 \), giving \( 12 \), and add this to \( -17 \).

```
4 |  3    -17    13    23
   |        12
   --------------------------
       3     -5
```

3. Multiply \( -5 \) by \( 4 \), giving \( -20 \), and add this to \( 13 \).

```
4 |  3    -17    13    23
   |        12   -20
   --------------------------
       3     -5    -7
```

4. Multiply \( -7 \) by \( 4 \), giving \( -28 \), and add this to \( 23 \).

```
4 |  3    -17    13    23
   |        12   -20   -28
   --------------------------
       3     -5    -7   -5
```

The quotient is \( 3x^2 - 5x - 7 \) and the remainder is \( -5 \).

### Step 3: Express the result.

We can now express the result in the required form:

\[
q(x) = 3x^2 - 5x - 7
\]
\[
r(x) = -5
\]
\[
b(x) = x - 4
\]

Thus, the final result is:

\[
3x^2 - 5x - 7 + \frac{-5}{x - 4}
\]

Do you have any questions or would you like more details on any step? 

Here are 5 related questions to expand your understanding:

1. How does synthetic division differ from long division in polynomials?
2. What would happen if we divided by \( x + 4 \) instead of \( x - 4 \)?
3. Can synthetic division be used when the divisor is not linear (e.g., \( x^2 + 2x + 1 \))?
4. How would the result change if there was no remainder?
5. Can the synthetic division method be applied to rational functions?

**Tip:** Synthetic division is much quicker than long division for dividing polynomials when the divisor is linear (in the form \( x - c \)).

Use synthetic division to find the result when 3x^3 - 17x^2 + 13x + 23 is divided by x - 4. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Learn how to use synthetic division to divide the polynomial 3x^3 - 17x^2 + 13x + 23 by x - 4. This detailed step-by-step solution helps you understand polynomial division and how to express the quotient and remainder. Suitable for high school students in Grades 10-12.

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