The GCD of `(2x^(2)-4x), (3x^(4)-12x^(2))` and `(2x^(5)-2x^(4)-4x^(3))` is :

Options

A. `x(x-2)`
B. `2x(x-2)`
C. `2x(2-x)`
D. `2x(2+x)`

Answer

To find the GCD (Greatest Common Divisor) of the polynomials \(2x^2 - 4x\), \(3x^4 - 12x^2\), and \(2x^5 - 2x^4 - 4x^3\), we will follow these steps:

### Step 1: Factor the first polynomial \(2x^2 - 4x\)
We can factor out the common term:
\[ 2x^2 - 4x = 2x(x - 2) \]

### Step 2: Factor the second polynomial \(3x^4 - 12x^2\)
We can factor out the common term:
\[ 3x^4 - 12x^2 = 3x^2(x^2 - 4) \]
Next, we can further factor \(x^2 - 4\) using the difference of squares:
\[ x^2 - 4 = (x - 2)(x + 2) \]
Thus, the complete factorization is:
\[ 3x^4 - 12x^2 = 3x^2(x - 2)(x + 2) \]

### Step 3: Factor the third polynomial \(2x^5 - 2x^4 - 4x^3\)
We can factor out the common term:
\[ 2x^5 - 2x^4 - 4x^3 = 2x^3(x^2 - x - 2) \]
Next, we can factor \(x^2 - x - 2\):
\[ x^2 - x - 2 = (x - 2)(x + 1) \]
Thus, the complete factorization is:
\[ 2x^5 - 2x^4 - 4x^3 = 2x^3(x - 2)(x + 1) \]

### Step 4: Write down the factorizations
Now we have:
1. \(2x^2 - 4x = 2x(x - 2)\)
2. \(3x^4 - 12x^2 = 3x^2(x - 2)(x + 2)\)
3. \(2x^5 - 2x^4 - 4x^3 = 2x^3(x - 2)(x + 1)\)

### Step 5: Identify the common factors
From the factorizations, we can see that:
- The common factor in all three expressions is \(x - 2\).
- The lowest power of \(x\) present in all three is \(x\) (from the first polynomial).

### Step 6: Combine the common factors
Thus, the GCD of the three polynomials is:
\[ \text{GCD} = x(x - 2) \]

### Final Answer
The GCD of \(2x^2 - 4x\), \(3x^4 - 12x^2\), and \(2x^5 - 2x^4 - 4x^3\) is:
\[ \boxed{x(x - 2)} \]