The non - repeated root of `x^3 +4x^2 +5x +2=0` is

Options

A. `-5/3`
B. `-2`
C. `-1`
D. `1`

Answer

To find the non-repeated root of the cubic equation \(x^3 + 4x^2 + 5x + 2 = 0\), we will follow these steps:

### Step 1: Identify the function
Let \(f(x) = x^3 + 4x^2 + 5x + 2\).

### Step 2: Test possible rational roots
Using the Rational Root Theorem, we can test possible rational roots. The possible rational roots can be the factors of the constant term (2) divided by the factors of the leading coefficient (1). Thus, the possible rational roots are \( \pm 1, \pm 2\).

### Step 3: Evaluate \(f(x)\) at the possible roots
Let's evaluate \(f(x)\) at \(x = -1\):
\[ f(-1) = (-1)^3 + 4(-1)^2 + 5(-1) + 2 = -1 + 4 - 5 + 2 = 0 \]
Since \(f(-1) = 0\), \(x = -1\) is a root.

### Step 4: Factor the polynomial
Since \(x = -1\) is a root, we can factor \(f(x)\) as:
\[ f(x) = (x + 1)(\text{quadratic factor}) \]
We will perform polynomial long division to find the quadratic factor.

### Step 5: Perform polynomial long division
Divide \(x^3 + 4x^2 + 5x + 2\) by \(x + 1\):
1. Divide \(x^3\) by \(x\) to get \(x^2\).
2. Multiply \(x^2\) by \(x + 1\) to get \(x^3 + x^2\).
3. Subtract:
\[ (x^3 + 4x^2 + 5x + 2) - (x^3 + x^2) = 3x^2 + 5x + 2 \]
4. Divide \(3x^2\) by \(x\) to get \(3x\).
5. Multiply \(3x\) by \(x + 1\) to get \(3x^2 + 3x\).
6. Subtract:
\[ (3x^2 + 5x + 2) - (3x^2 + 3x) = 2x + 2 \]
7. Divide \(2x\) by \(x\) to get \(2\).
8. Multiply \(2\) by \(x + 1\) to get \(2x + 2\).
9. Subtract:
\[ (2x + 2) - (2x + 2) = 0 \]

Thus, we have:
\[ f(x) = (x + 1)(x^2 + 3x + 2) \]

### Step 6: Factor the quadratic
Now we need to factor \(x^2 + 3x + 2\):
\[ x^2 + 3x + 2 = (x + 1)(x + 2) \]

### Step 7: Write the complete factorization
Now we can write:
\[ f(x) = (x + 1)(x + 1)(x + 2) = (x + 1)^2(x + 2) \]

### Step 8: Identify the roots
From the factorization, the roots are:
- \(x = -1\) (repeated root)
- \(x = -2\) (non-repeated root)

### Conclusion
The non-repeated root of the equation \(x^3 + 4x^2 + 5x + 2 = 0\) is:
\[ \boxed{-2} \]