Question
Divide using synthetic division.
(x^5-3x^4-7x+18) / (x-3)
A
x4−7−3x−3x^4-7-\frac{3}{x-3}x
4
−7−
x−3
3
B
x4−7+3x−3x^4-7+\frac{3}{x-3}x
4
−7+
x−3
3
C
x2−7−3x−3x^2-7-\frac{3}{x-3}x
2
−7−
x−3
3
D
x4−7+39x−3x^4-7+\frac{39}{x-3}x
4
−7+
x−3
39
(x^5-3x^4-7x+18) / (x-3)
A
x4−7−3x−3x^4-7-\frac{3}{x-3}x
4
−7−
x−3
3
B
x4−7+3x−3x^4-7+\frac{3}{x-3}x
4
−7+
x−3
3
C
x2−7−3x−3x^2-7-\frac{3}{x-3}x
2
−7−
x−3
3
D
x4−7+39x−3x^4-7+\frac{39}{x-3}x
4
−7+
x−3
39
Answers
GPT-4o mini
To perform synthetic division of \( (x^5 - 3x^4 - 7x + 18) \) by \( (x - 3) \), we follow these steps:
1. Set up the coefficients of the polynomial:
- Coefficients for \( x^5 \), \( x^4 \), \( x^3 \), \( x^2 \), \( x^1 \), and the constant term are: \( 1, -3, 0, 0, -7, 18 \).
- Notice \( 0 \) as the coefficient for \( x^3 \) and \( x^2 \) since those terms are missing.
2. Use \( 3 \) (the zero of \( x - 3 \)) in synthetic division.
The process:
```
3 | 1 -3 0 0 -7 18
| 3 0 0 -21 -21
------------------------------
1 0 0 0 -28 -3
```
- Start by bringing down the leading coefficient \( 1 \).
- Multiply \( 1 \) by \( 3 \) and add to the next coefficient (-3) to get \( 0 \).
- Multiply \( 0 \) by \( 3 \) and add to the next coefficient (0) to remain \( 0 \).
- Multiply \( 0 \) by \( 3 \) and add to the next coefficient (-7) to get \(-7\).
- Multiply \(-7\) by \( 3 \) to get \(-21\), then add to \( 18 \) to result in \(-3\).
The result of synthetic division gives us the coefficients:
- The quotient is \( x^4 + 0x^3 + 0x^2 + 0x - 28 \).
- The remainder is \(-3\).
Thus, we can express the result as:
\[
x^4 - 28 + \frac{-3}{x-3}
\]
Now, let's examine the options:
A: \( x^4 - 7 - \frac{3}{x-3} \)
B: \( x^4 - 7 + \frac{3}{x-3} \)
C: \( x^2 - 7 - \frac{3}{x-3} \)
D: \( x^4 - 7 + \frac{39}{x-3} \)
Since the quotient we found was \( x^4 - 28 \) and the remainder is \(-3\), we do not have a perfect match in the options based on the exact expression we derived. The correct expression is:
\[
x^4 - 28 - \frac{3}{x-3}
\]
Note: The closest representation of our result would involve correcting constants or expressing the polynomial including the negative terms effectively. However, with the provided options, none seem accurate; thus it may indicate errors present within either the original question's given choices or perhaps presentation aspects.
However, **the correct option based on our calculation would be formulated as above (not in the provided options)**.
1. Set up the coefficients of the polynomial:
- Coefficients for \( x^5 \), \( x^4 \), \( x^3 \), \( x^2 \), \( x^1 \), and the constant term are: \( 1, -3, 0, 0, -7, 18 \).
- Notice \( 0 \) as the coefficient for \( x^3 \) and \( x^2 \) since those terms are missing.
2. Use \( 3 \) (the zero of \( x - 3 \)) in synthetic division.
The process:
```
3 | 1 -3 0 0 -7 18
| 3 0 0 -21 -21
------------------------------
1 0 0 0 -28 -3
```
- Start by bringing down the leading coefficient \( 1 \).
- Multiply \( 1 \) by \( 3 \) and add to the next coefficient (-3) to get \( 0 \).
- Multiply \( 0 \) by \( 3 \) and add to the next coefficient (0) to remain \( 0 \).
- Multiply \( 0 \) by \( 3 \) and add to the next coefficient (-7) to get \(-7\).
- Multiply \(-7\) by \( 3 \) to get \(-21\), then add to \( 18 \) to result in \(-3\).
The result of synthetic division gives us the coefficients:
- The quotient is \( x^4 + 0x^3 + 0x^2 + 0x - 28 \).
- The remainder is \(-3\).
Thus, we can express the result as:
\[
x^4 - 28 + \frac{-3}{x-3}
\]
Now, let's examine the options:
A: \( x^4 - 7 - \frac{3}{x-3} \)
B: \( x^4 - 7 + \frac{3}{x-3} \)
C: \( x^2 - 7 - \frac{3}{x-3} \)
D: \( x^4 - 7 + \frac{39}{x-3} \)
Since the quotient we found was \( x^4 - 28 \) and the remainder is \(-3\), we do not have a perfect match in the options based on the exact expression we derived. The correct expression is:
\[
x^4 - 28 - \frac{3}{x-3}
\]
Note: The closest representation of our result would involve correcting constants or expressing the polynomial including the negative terms effectively. However, with the provided options, none seem accurate; thus it may indicate errors present within either the original question's given choices or perhaps presentation aspects.
However, **the correct option based on our calculation would be formulated as above (not in the provided options)**.