To divide the polynomial [tex]\( x^3 - 2x^2 + 16 \)[/tex] by [tex]\( x + 2 \)[/tex], we can use polynomial long division. Let me guide you through the long division process step by step.
1. Setup the Division:
Write the dividend [tex]\( x^3 - 2x^2 + 16 \)[/tex] under the long division symbol and the divisor [tex]\( x + 2 \)[/tex] outside.
```
x^2 - 4x + 8
___________________________________________________
x + 2 | x^3 - 2x^2 + 0x + 16
```
2. First Division:
- Divide the leading term of the dividend [tex]\( x^3 \)[/tex] by the leading term of the divisor [tex]\( x \)[/tex], which gives [tex]\( x^2 \)[/tex].
- Multiply the entire divisor [tex]\( x + 2 \)[/tex] by [tex]\( x^2 \)[/tex] and subtract from the original polynomial.
[tex]\[
x^3 - 2x^2 + 0x + 16
- (x^3 + 2x^2)
= -4x^2 + 0x + 16
```
3. Second Division:
- Divide the new dividend's leading term \( -4x^2 \) by the leading term of the divisor \( x \), which gives \( -4x \).
- Multiply the entire divisor \( x + 2 \) by \( -4x \) and subtract from the current result.
\[
-4x^2 + 0x + 16
- (-4x^2 - 8x)
= 8x + 16
```
4. Third Division:
- Divide the new dividend's leading term \( 8x \) by the leading term of the divisor \( x \), which gives \( 8 \).
- Multiply the entire divisor \( x + 2 \) by \( 8 \) and subtract from the current result.
\[
8x + 16
- (8x + 16)
= 0
```
After performing these steps, we arrive at:
Quotient: \( x^2 - 4x + 8 \) \
Remainder: \( 0 \)
Therefore, the result of dividing \( x^3 - 2x^2 + 16 \) by \( x + 2 \) is:
\[ \boxed{(x^2 - 4x + 8, 0)} \][/tex]
