Which of the following is a factor of x^3 - 5x^2 - 2x + 24, given that x - 3 is one of the linear factors?
If x - 3 is a factor of x^3 - 5x^2 - 2x + 24, that means that (x - 3) divides evenly into x^3 - 5x^2 - 2x + 24 when we perform long division.
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x - 3 | x^3 - 5x^2 - 2x + 24
To find the other factor, we perform long division:
x^2 - 2x - 6
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x - 3 | x^3 - 5x^2 - 2x + 24
- (x^3 - 3x^2)
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-2x^2 - 2x
+ (2x^2 - 6x)
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4x + 24
- (4x + 12)
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12
The remainder is 12. Thus, the other factor is x^2 - 2x - 6, and the three factors of x^3 - 5x^2 - 2x + 24 are (x - 3)(x^2 - 2x - 6) + 12.
Therefore, the correct answer is x^2 - 2x - 6.
If x - 3 is one of the linear factors of the polynomial x^3 - 5x^2 - 2x + 24, we can use synthetic division to find the remaining quadratic factor.
Using synthetic division with the given linear factor x - 3:
3 │ 1 -5 -2 24
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1 -2 -8 0
The result of the synthetic division is 1x^2 - 2x - 8, which represents the quadratic factor. Therefore, x^3 - 5x^2 - 2x + 24 can be factored as (x - 3)(x^2 - 2x - 8).
To find the other factor, we can use polynomial long division or synthetic division. Let's use synthetic division to divide x^3 - 5x^2 - 2x + 24 by x - 3.
Step 1: Write down the coefficients of the polynomial in descending order of the powers of x.
1 -5 -2 24
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Step 2: Draw a line and write down the divisor, x - 3, on the left.
x - 3
Step 3: Bring down the first coefficient, which is 1, below the line.
x - 3
1
Step 4: Multiply the divisor (x - 3) by the first term (1) and write the result above the line, directly below the second coefficient (-5).
x - 3
1
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x - 3
Step 5: Subtract the product from the second term (-5) and write the result below the line.
x - 3
1
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x - 3
-5
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4
Step 6: Repeat steps 4 and 5 until all terms have been processed.
x - 3
1 4
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x - 3
-5 4
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1
The result of the division is x - 3 with a remainder of 1. This means that (x - 3) is a factor of x^3 - 5x^2 - 2x + 24.
So, the other factor is x - 3.