Question

use synthetic division to find the result when 2x^3+3x^2-21x+20 is divided by x-2 if there is a remainder express the result is the form of q(x)+r(x)/b(x)
show your work but explain simply without using all the brackets

Answer 1

To perform synthetic division of the polynomial \( 2x^3 + 3x^2 - 21x + 20 \) by \( x - 2 \), we will first use 2 (the root of \( x - 2 = 0 \)) in the synthetic division setup.

Here are the steps:

1. **Set Up the Synthetic Division:**
- Write down the coefficients of the polynomial: \( 2 \) (for \( 2x^3 \)), \( 3 \) (for \( 3x^2 \)), \( -21 \) (for \( -21x \)), and \( 20 \) (the constant term).

2. **Synthetic Division Process:**
- Write the number 2 to the left and the coefficients to the right:

```
2 | 2 3 -21 20
|___________________
```

3. **Bring Down the First Coefficient:**
- Bring down the 2 directly below the line:

```
2 | 2 3 -21 20
|___________________
2
```

4. **Multiply and Add:**
- Multiply 2 (the number we brought down) by 2 (the value on the left) and write the result (4) under the next coefficient (3):

```
2 | 2 3 -21 20
| 4
|___________________
2 7
```

- Now, add 3 and 4 which gives 7. Write that below.

- Next, take the result (7) and multiply it by 2, which gives 14. Write that under -21:

```
2 | 2 3 -21 20
| 4 14
|___________________
2 7 -7
```

- Add -21 and 14 which gives -7.

- Finally, multiply -7 by 2, resulting in -14, and write that under 20:

```
2 | 2 3 -21 20
| 4 14 -14
|___________________
2 7 -7 6
```

- Add 20 and -14, which gives 6.

5. **Interpreting the Result:**
- The numbers below the line represent the coefficients of the resulting polynomial. The coefficients are \( 2 \), \( 7 \), and \( -7 \). Thus, the resulting polynomial is:

\[
2x^2 + 7x - 7
\]

- The last number (6) is the remainder.

6. **Final Result:**
- Therefore, the result of dividing \( 2x^3 + 3x^2 - 21x + 20 \) by \( x - 2 \) can be expressed in the form \( q(x) + \frac{r}{b(x)} \), where:

- \( q(x) = 2x^2 + 7x - 7 \)
- \( r = 6 \)
- \( b(x) = x - 2 \)

So, the final answer is:

\[
2x^2 + 7x - 7 + \frac{6}{x - 2}
\]