Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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]
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]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.