Sure, let's work through the polynomial division of [tex]\( \frac{x^3 - 1}{x - 1} \)[/tex].
To divide [tex]\( x^3 - 1 \)[/tex] by [tex]\( x - 1 \)[/tex], we can use polynomial long division. Here are the steps:
1. Set up the division: Write [tex]\( x^3 - 1 \)[/tex] as [tex]\( x^3 + 0x^2 + 0x - 1 \)[/tex] to account for all terms.
```plaintext
_______________
x - 1 | x^3 + 0x^2 + 0x - 1
```
2. Divide the leading term: Divide the leading term of the dividend [tex]\( x^3 \)[/tex] by the leading term of the divisor [tex]\( x \)[/tex]. This gives us [tex]\( x^2 \)[/tex].
```plaintext
x^2
_______________
x - 1 | x^3 + 0x^2 + 0x - 1
```
3. Multiply and subtract: Multiply [tex]\( x^2 \)[/tex] by the entire divisor [tex]\( x - 1 \)[/tex], and subtract this product from the original dividend.
```plaintext
x^2
_______________
x - 1 | x^3 + 0x^2 + 0x - 1
-(x^3 - x^2)
_______________
x^2 + 0x - 1
```
4. Bring down the next term: The result is [tex]\( x^2 + 0x - 1 \)[/tex].
5. Repeat the process: Now, divide the leading term [tex]\( x^2 \)[/tex] by [tex]\( x \)[/tex], which gives [tex]\( x \)[/tex]. Then multiply [tex]\( x \)[/tex] by [tex]\( x - 1 \)[/tex] and subtract again.
```plaintext
x^2 + x
_______________
x - 1 | x^3 + 0x^2 + 0x - 1
-(x^3 - x^2)
_______________
x^2 + 0x - 1
-(x^2 - x)
_______________
x - 1
```
6. Divide again: Divide the leading term [tex]\( x \)[/tex] by [tex]\( x \)[/tex], which gives 1. Multiply 1 by [tex]\( x - 1 \)[/tex] and subtract.
```plaintext
x^2 + x + 1
_______________
x - 1 | x^3 + 0x^2 + 0x - 1
-(x^3 - x^2)
_______________
x^2 + 0x - 1
-(x^2 - x)
_______________
x - 1
-(x - 1)
_______________
0
```
The remainder is 0, and the quotient is [tex]\( x^2 + x + 1 \)[/tex].
Therefore, the result of the polynomial division [tex]\( \frac{x^3 - 1}{x - 1} \)[/tex] is:
[tex]\[ x^2 + x + 1 \][/tex]
