Question
Divide the following using synthetic division.
(
x
3
−
4
)
÷
(
x
+
2
)
(x
3
−4)÷(x+2)
(
x
3
−
4
)
÷
(
x
+
2
)
(x
3
−4)÷(x+2)
Answers
GPT-4o mini
To perform synthetic division of the polynomial \( x^3 - 4 \) by \( x + 2 \), we first note that synthetic division requires us to use the root of the divisor. The polynomial \( x + 2 \) has a root at \( x = -2 \).
Here are the steps to perform synthetic division:
1. Identify the coefficients of the polynomial \( x^3 - 4 \). Since it's a cubic polynomial, we write it as \( 1x^3 + 0x^2 + 0x - 4 \). The coefficients are:
- 1 (for \( x^3 \))
- 0 (for \( x^2 \))
- 0 (for \( x^1 \))
- -4 (for the constant term)
So the coefficients are: \( [1, 0, 0, -4] \).
2. Set up the synthetic division using the root \( -2 \):
```
-2 | 1 0 0 -4
|
|________________
```
3. Bring down the leading coefficient:
```
-2 | 1 0 0 -4
|
|________________
1
```
4. Multiply \( -2 \) by the value just brought down (which is 1) and write the result under the next coefficient:
```
-2 | 1 0 0 -4
| -2
|________________
1 -2
```
5. Now, add the values in the second column:
```
-2 | 1 0 0 -4
| -2
|________________
1 -2 0
```
6. Repeat this process for the next column. Multiply \( -2 \) by \( -2 \) (which gives \( 4 \)) and write it under the next coefficient (which is 0):
```
-2 | 1 0 0 -4
| -2 4
|________________
1 -2 4
```
7. Now add the values in the third column:
```
-2 | 1 0 0 -4
| -2 4
|________________
1 -2 4 0
```
8. Finally, multiply \( -2 \) by \( 4 \) (which gives \( -8 \)) and add that to the last coefficient:
```
-2 | 1 0 0 -4
| -2 4 -8
|________________
1 -2 4 -12
```
The result of the synthetic division is the polynomial \( x^2 - 2x + 4 \) plus a remainder of \( -12 \). Putting this all together, we can express the result of the division as:
\[
x^3 - 4 = (x + 2)(x^2 - 2x + 4) - 12
\]
So, in summary, the result of the synthetic division is:
\[
x^3 - 4 = (x + 2)(x^2 - 2x + 4) - 12
\]
Thus, \( (x^3 - 4) \div (x + 2) = x^2 - 2x + 4 \) with a remainder of \( -12 \).
Here are the steps to perform synthetic division:
1. Identify the coefficients of the polynomial \( x^3 - 4 \). Since it's a cubic polynomial, we write it as \( 1x^3 + 0x^2 + 0x - 4 \). The coefficients are:
- 1 (for \( x^3 \))
- 0 (for \( x^2 \))
- 0 (for \( x^1 \))
- -4 (for the constant term)
So the coefficients are: \( [1, 0, 0, -4] \).
2. Set up the synthetic division using the root \( -2 \):
```
-2 | 1 0 0 -4
|
|________________
```
3. Bring down the leading coefficient:
```
-2 | 1 0 0 -4
|
|________________
1
```
4. Multiply \( -2 \) by the value just brought down (which is 1) and write the result under the next coefficient:
```
-2 | 1 0 0 -4
| -2
|________________
1 -2
```
5. Now, add the values in the second column:
```
-2 | 1 0 0 -4
| -2
|________________
1 -2 0
```
6. Repeat this process for the next column. Multiply \( -2 \) by \( -2 \) (which gives \( 4 \)) and write it under the next coefficient (which is 0):
```
-2 | 1 0 0 -4
| -2 4
|________________
1 -2 4
```
7. Now add the values in the third column:
```
-2 | 1 0 0 -4
| -2 4
|________________
1 -2 4 0
```
8. Finally, multiply \( -2 \) by \( 4 \) (which gives \( -8 \)) and add that to the last coefficient:
```
-2 | 1 0 0 -4
| -2 4 -8
|________________
1 -2 4 -12
```
The result of the synthetic division is the polynomial \( x^2 - 2x + 4 \) plus a remainder of \( -12 \). Putting this all together, we can express the result of the division as:
\[
x^3 - 4 = (x + 2)(x^2 - 2x + 4) - 12
\]
So, in summary, the result of the synthetic division is:
\[
x^3 - 4 = (x + 2)(x^2 - 2x + 4) - 12
\]
Thus, \( (x^3 - 4) \div (x + 2) = x^2 - 2x + 4 \) with a remainder of \( -12 \).