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 remaining factor, we can use polynomial long division.

x^2 + 6x + 8
______________________
x - 3 | x^3 - 5x^2 - 2x + 24
x^3 - 3x^2
_______________________
- 2x^2 - 2x
- 2x^2 + 6x
__________________
4x + 24
4x - 12
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36

So, x^3 - 5x^2 - 2x + 24 = (x - 3)(x^2 + 6x + 8) + 36

Therefore, the remaining factor is 36.

Among the given options, none of them are equal to 36. Thus, none of the given options is a factor of x^3 - 5x^2 - 2x + 24.

To find the factor of the given polynomial, we will use polynomial division.

Step 1: Set up the long division:
-------------
(x - 3) | x^3 - 5x^2 - 2x + 24

Step 2: Divide the first term of the polynomial by the first term of the divisor:
x^3 ÷ x = x^2

Step 3: Multiply the result obtained in Step 2 by the entire divisor:
x^2 * (x - 3) = x^3 - 3x^2

Step 4: Subtract the result obtained in Step 3 from the original polynomial:
(x^3 - 5x^2 - 2x + 24) - (x^3 - 3x^2) = -2x^2 - 2x + 24

Step 5: Repeat steps 2-4 with the new polynomial obtained:
-2x^2 ÷ x = -2x

-2x * (x - 3) = -2x^2 + 6x

(-2x^2 - 2x + 24) - (-2x^2 + 6x) = -8x + 24

Step 6: Repeat Steps 2-4 again:
-8x ÷ x = -8

-8 * (x - 3) = -8x + 24

(-8x + 24) - (-8x + 24) = 0

The resulting polynomial after all the divisions is 0. This means that (x - 3) is a factor of x^3 - 5x^2 - 2x + 24.

Now let's see which of the given options are also factors:

a) For x - 2:
We can substitute x = 2 into the polynomial and check if it equals zero:
(2)^3 - 5(2)^2 - 2(2) + 24 = 8 - 20 - 4 + 24 = 8 - 24 + 24 = 8

Since the result is not zero, x - 2 is not a factor.

b) For x + 4:
We can substitute x = -4 into the polynomial and check if it equals zero:
(-4)^3 - 5(-4)^2 - 2(-4) + 24 = -64 - 80 + 8 + 24 = -144 + 32 = -112

Since the result is not zero, x + 4 is not a factor.

c) For x^2 - 8x + 22:
We can substitute this expression into the polynomial and check if it equals zero:
(x^2 - 8x + 22)(x - 3) = (x^3 - 3x^2 - 8x^2 + 24x + 22x - 66) = x^3 - 11x^2 + 46x - 66

Since this is not equal to the original polynomial x^3 - 5x^2 - 2x + 24, x^2 - 8x + 22 is not a factor.

d) For x + 2:
We can substitute x = -2 into the polynomial and check if it equals zero:
(-2)^3 - 5(-2)^2 - 2(-2) + 24 = -8 - 20 + 4 + 24 = -28 + 28 = 0

Since the result is zero, x + 2 is a factor.

Therefore, the factor of x^3 - 5x^2 - 2x + 24, given that x - 3 is a factor, is x + 2.