Question
Use long division to divide.
3k2−18k−463x+6
(1 point)
Responses
k−8
k minus 8
k−8+23k+6
k minus 8 plus the fraction with numerator 2 and denominator 3 k plus 6
k−8−943k+6
k minus 8 minus the fraction with numerator 94 and denominator 3 k plus 6
k−8+13k+3
3k2−18k−463x+6
(1 point)
Responses
k−8
k minus 8
k−8+23k+6
k minus 8 plus the fraction with numerator 2 and denominator 3 k plus 6
k−8−943k+6
k minus 8 minus the fraction with numerator 94 and denominator 3 k plus 6
k−8+13k+3
Answers
Answered by
GPT-4o mini
To divide the polynomial \( 3k^2 - 18k - 463 \) by \( k - 8 \) using long division, follow these steps:
1. **Set up the long division**: Write \( 3k^2 - 18k - 463 \) under the long division symbol and \( k - 8 \) outside.
2. **Divide the first term**: Divide the leading term \( 3k^2 \) by \( k \). This gives us \( 3k \).
3. **Multiply**: Multiply \( 3k \) by \( k - 8 \):
\[
3k \cdot (k - 8) = 3k^2 - 24k
\]
4. **Subtract**: Now subtract \( 3k^2 - 24k \) from \( 3k^2 - 18k - 463 \):
\[
(3k^2 - 18k - 463) - (3k^2 - 24k) = 6k - 463
\]
5. **Bring down the next term**: The next term to consider, in this case, is \( 6 \), so we rewrite it as \( 6k - 463 + 6 \).
6. **Divide the leading term again**: Now divide \( 6k \) by \( k \), which gives us \( 6 \).
7. **Multiply again**: Multiply \( 6 \) by \( k - 8 \):
\[
6 \cdot (k - 8) = 6k - 48
\]
8. **Subtract again**: Now subtract \( 6k - 48 \) from \( 6k - 463 \):
\[
(6k - 463) - (6k - 48) = -463 + 48 = -415
\]
At this point, we cannot divide \( -415 \) by \( k - 8 \) since the degree of the remainder is lower than the degree of the divisor.
### Final Result:
Combining all parts, we get:
\[
3k + 6 - \frac{415}{k - 8}
\]
This type of mixed expression can be summarized as:
- \( k - 8 + \frac{number}{denominator} \)
From the given options, you want to look beyond the initial computations and focus on the constant term's relationship with expressions, particularly recognizing the outcomes effectively describe their respective results.
However, in this situation, the best relevant choice given the options would likely be:
- **\( k - 8 - \frac{94}{3k + 6} \)**
But based on the actual division outcome, all provided responses do not match precisely the accurate division outcome of \( 3k + 6 - \frac{415}{k - 8} \). Thus the pristine correct answer hasn’t been demonstrated among the options available.
Feel free to utilize this understanding or provide an appropriate correction or precise collation.
1. **Set up the long division**: Write \( 3k^2 - 18k - 463 \) under the long division symbol and \( k - 8 \) outside.
2. **Divide the first term**: Divide the leading term \( 3k^2 \) by \( k \). This gives us \( 3k \).
3. **Multiply**: Multiply \( 3k \) by \( k - 8 \):
\[
3k \cdot (k - 8) = 3k^2 - 24k
\]
4. **Subtract**: Now subtract \( 3k^2 - 24k \) from \( 3k^2 - 18k - 463 \):
\[
(3k^2 - 18k - 463) - (3k^2 - 24k) = 6k - 463
\]
5. **Bring down the next term**: The next term to consider, in this case, is \( 6 \), so we rewrite it as \( 6k - 463 + 6 \).
6. **Divide the leading term again**: Now divide \( 6k \) by \( k \), which gives us \( 6 \).
7. **Multiply again**: Multiply \( 6 \) by \( k - 8 \):
\[
6 \cdot (k - 8) = 6k - 48
\]
8. **Subtract again**: Now subtract \( 6k - 48 \) from \( 6k - 463 \):
\[
(6k - 463) - (6k - 48) = -463 + 48 = -415
\]
At this point, we cannot divide \( -415 \) by \( k - 8 \) since the degree of the remainder is lower than the degree of the divisor.
### Final Result:
Combining all parts, we get:
\[
3k + 6 - \frac{415}{k - 8}
\]
This type of mixed expression can be summarized as:
- \( k - 8 + \frac{number}{denominator} \)
From the given options, you want to look beyond the initial computations and focus on the constant term's relationship with expressions, particularly recognizing the outcomes effectively describe their respective results.
However, in this situation, the best relevant choice given the options would likely be:
- **\( k - 8 - \frac{94}{3k + 6} \)**
But based on the actual division outcome, all provided responses do not match precisely the accurate division outcome of \( 3k + 6 - \frac{415}{k - 8} \). Thus the pristine correct answer hasn’t been demonstrated among the options available.
Feel free to utilize this understanding or provide an appropriate correction or precise collation.