Which of the following is a factor of x^3−5x^2−2x+24, given that x−3 is one of the linear factors?
x-2
x+4
x^2-8x+22
x+2
In order to find the other factor, we can use polynomial long division. We divide x^3 - 5x^2 - 2x + 24 by x - 3:
x^2 - 2x - 8
______________________
x - 3 | x^3 - 5x^2 - 2x + 24
- (x^3 - 3x^2)
_______________
- 2x^2 - 2x
+ (2x^2 - 6x)
_______________
- 4x + 24
- (-4x + 12)
_______________
12
The remainder is 12, which means that x - 3 is a factor of x^3 - 5x^2 - 2x + 24.
From the division, we obtain the quotient as x^2 - 2x - 8.
Thus, the other factor is x^2 - 2x - 8.
None of the given options (x-2, x+4, x^2-8x+22, x+2) are the correct factor. The correct factor is x^2 - 2x - 8.
To determine which of the options is a factor of \(x^3 - 5x^2 - 2x + 24\) given that \(x - 3\) is one of the linear factors, you can use synthetic division.
First, write the coefficients of the polynomial: 1, -5, -2, 24. Then use the value 3, since \(x - 3\) is a linear factor.
Perform synthetic division:
```
3 | 1 -5 -2 24
| 3 -6 12
------------------
1 -2 10 36
```
The remainder of the division is 36, not zero.
Therefore, \(x - 3\) is not a factor of \(x^3 - 5x^2 - 2x + 24\).
Now, let's check the other options:
a) \(x - 2\):
Perform synthetic division with 2:
```
2 | 1 -5 -2 24
| 2 -6 -16
------------------
1 -3 -8 8
```
The remainder is 8, so \(x - 2\) is not a factor.
b) \(x + 4\):
Perform synthetic division with -4:
```
-4 | 1 -5 -2 24
| -4 36 -128
-------------------
1 -9 34 -104
```
The remainder is -104, so \(x + 4\) is not a factor.
c) \(x^2 - 8x + 22\):
Perform synthetic division with -3:
```
-3 | 1 -5 -2 24
| -3 24 -66
-------------------
1 -8 22 -42
```
The remainder is -42, so \(x^2 - 8x + 22\) is not a factor.
d) \(x + 2\):
Perform synthetic division with -2:
```
-2 | 1 -5 -2 24
| -2 14 -24
-------------------
1 -7 12 0
```
The remainder is 0, so \(x + 2\) is a factor of \(x^3 - 5x^2 - 2x + 24\).
Therefore, the correct answer is \(x + 2\).