What is the LCM of `X^2 + 2X-8 , X^3-4X^2 + 4X and x^2 + 4x` ?

Options

A. `x(x+4) (x-2)^2`
B. `x(x+4)(x-2)`
C. `x(x+4) (x+2)^2`
D. `x(x+4)^2 (x-2)`

Answer

To find the LCM of the polynomials \(X^2 + 2X - 8\), \(X^3 - 4X^2 + 4X\), and \(X^2 + 4X\), we will follow these steps:

### Step 1: Factorize each polynomial

1. **Factorizing \(X^2 + 2X - 8\)**:
- We need to find two numbers that add up to 2 and multiply to -8. The numbers are 4 and -2.
- Rewrite the polynomial:
\[ X^2 + 4X - 2X - 8 = (X^2 + 4X) + (-2X - 8) \]
- Factor by grouping:
\[ X(X + 4) - 2(X + 4) = (X - 2)(X + 4) \]

2. **Factorizing \(X^3 - 4X^2 + 4X\)**:
- First, factor out the common term \(X\):
\[ X(X^2 - 4X + 4) \]
- Now, factor the quadratic \(X^2 - 4X + 4\). This can be factored as:
\[ (X - 2)(X - 2) = (X - 2)^2 \]
- Therefore, we have:
\[ X(X - 2)^2 \]

3. **Factorizing \(X^2 + 4X\)**:
- Factor out the common term \(X\):
\[ X(X + 4) \]

### Step 2: Write down the factorizations

Now we have the factorizations:
- \(X^2 + 2X - 8 = (X - 2)(X + 4)\)
- \(X^3 - 4X^2 + 4X = X(X - 2)^2\)
- \(X^2 + 4X = X(X + 4)\)

### Step 3: Determine the LCM

To find the LCM, we take the highest power of each factor from the factorizations:

- From \(X^2 + 2X - 8\): \(X - 2\) (power 1), \(X + 4\) (power 1)
- From \(X^3 - 4X^2 + 4X\): \(X\) (power 1), \(X - 2\) (power 2)
- From \(X^2 + 4X\): \(X\) (power 1), \(X + 4\) (power 1)

Combining these, we get:
- \(X\) (highest power is 1)
- \(X - 2\) (highest power is 2)
- \(X + 4\) (highest power is 1)

Thus, the LCM is:
\[ LCM = X \cdot (X - 2)^2 \cdot (X + 4) \]

### Final Answer

The LCM of the given polynomials is:
\[ X(X - 2)^2(X + 4) \]