The roots of the equation `x^3 -3x -2=0` are

Options

A. `-1,-1,2`
B. `-1,1,-2`
C. `-1,2-3`
D. `-1,-1,-2`

Answer

To find the roots of the equation \(x^3 - 3x - 2 = 0\), we can follow these steps:

### Step 1: Identify possible rational roots using the Rational Root Theorem.
The Rational Root Theorem suggests that any rational solution, in the form of \( \frac{p}{q} \), where \( p \) is a factor of the constant term (-2) and \( q \) is a factor of the leading coefficient (1). The factors of -2 are ±1, ±2.

**Hint:** Check the possible rational roots by substituting them into the equation.

### Step 2: Test the possible rational roots.
Let's test \( x = -1 \):

\[ (-1)^3 - 3(-1) - 2 = -1 + 3 - 2 = 0 \]

Since \( x = -1 \) is a root, we can factor the polynomial.

**Hint:** If a value satisfies the equation, it is a root.

### Step 3: Factor the polynomial using synthetic division.
Now that we know \( x + 1 \) is a factor, we can perform synthetic division of \( x^3 - 3x - 2 \) by \( x + 1 \).

1. Write the coefficients: 1 (for \( x^3 \)), 0 (for \( x^2 \)), -3 (for \( x \)), -2 (constant).
2. Use -1 in synthetic division:

```
-1 | 1 0 -3 -2
| -1 1 2
---------------------
1 -1 -2 0
```

The result is \( x^2 - x - 2 \).

**Hint:** Synthetic division helps simplify the polynomial to a quadratic.

### Step 4: Factor the quadratic.
Now we need to factor \( x^2 - x - 2 \):

\[ x^2 - x - 2 = (x - 2)(x + 1) \]

**Hint:** Look for two numbers that multiply to -2 and add to -1.

### Step 5: Write the complete factorization.
Now we can write the complete factorization of the original polynomial:

\[ x^3 - 3x - 2 = (x + 1)(x - 2)(x + 1) \]

**Hint:** Repeating factors indicate multiple roots.

### Step 6: Find the roots.
Setting each factor to zero gives us the roots:

1. \( x + 1 = 0 \) → \( x = -1 \) (double root)
2. \( x - 2 = 0 \) → \( x = 2 \)

Thus, the roots of the equation \( x^3 - 3x - 2 = 0 \) are:

\[ x = -1, -1, 2 \]

**Final Answer:** The roots are \( -1, -1, 2 \).

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