Answer:
C = -6
Step by step explanation:
If x + 2 is a factor of 3x³ + cx² - 3x - 2, it means that when we divide 3x³ + cx² - 3x - 2 by x + 2, the remainder is 0. We can use polynomial long division to find the value of c that satisfies this condition.
The division looks like:
3x^2 - 6x + c
x + 2 | 3x^3 + cx^2 - 3x - 2
- (3x^3 + 6x^2)
---------------
-5x^2 - 3x
+(-5x^2 - 10x)
-------------
7x - 2
Since the remainder is not zero, we need to add a term to the dividend to get a zero remainder. We can add the term 5x + 10 to the dividend, which gives:
3x^2 - x + c - 1
x + 2 | 3x^3 + cx^2 - 3x - 2 + 5x + 10
- (3x^3 + 6x^2)
---------------
-5x^2 - 3x
+(-5x^2 - 10x)
-------------
7x - 2 + 10
--------
7x + 8
Now, the remainder is 7x + 8. For x + 2 to be a factor, the remainder must be 0. Therefore, we need to choose c such that 7x + 8 is divisible by x + 2. We can do this by setting x = -2 and solving for c:
7(-2) + 8 = -14 + 8 = -6
Therefore, we need to choose c = -6 so that x + 2 is a factor of 3x³ + cx² - 3x - 2