Find the number of solution of the following equation `x^(4)-6x^(2)-8x-3=0`

Answer

To find the number of solutions for the equation \( x^4 - 6x^2 - 8x - 3 = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ x^4 - 6x^2 - 8x - 3 = 0 \]

### Step 2: Factor the polynomial
We will try to factor the polynomial. Notice that we can group terms to help with factoring:
\[ x^4 - 6x^2 - 8x - 3 = (x^4 - 6x^2) + (-8x - 3) \]

### Step 3: Use substitution
Let’s use substitution to simplify the polynomial. Let \( y = x^2 \). Then we have:
\[ y^2 - 6y - 8x - 3 = 0 \]
However, this does not seem to simplify directly. Instead, let's try to find roots directly.

### Step 4: Identify potential rational roots
Using the Rational Root Theorem, we can test for possible rational roots. The possible rational roots are factors of \(-3\) (the constant term) divided by factors of \(1\) (the leading coefficient):
Possible roots: \( \pm 1, \pm 3 \).

### Step 5: Test potential roots
We will test \( x = 3 \):
\[ 3^4 - 6(3^2) - 8(3) - 3 = 81 - 54 - 24 - 3 = 0 \]
Thus, \( x = 3 \) is a root.

### Step 6: Factor out \( (x - 3) \)
Now we can factor \( (x - 3) \) out of the polynomial. We can perform synthetic division or polynomial long division:
\[ x^4 - 6x^2 - 8x - 3 = (x - 3)(x^3 + 3x^2 + 1) \]

### Step 7: Solve the cubic equation
Now we need to find the roots of the cubic equation \( x^3 + 3x^2 + 1 = 0 \). We can use the discriminant or numerical methods to find the number of real roots.

### Step 8: Analyze the cubic equation
To find the number of real roots of the cubic equation, we can check the derivative:
\[ f'(x) = 3x^2 + 6x \]
Setting \( f'(x) = 0 \):
\[ 3x(x + 2) = 0 \implies x = 0 \text{ or } x = -2 \]
Now we check the values of \( f(x) \) at these critical points and at the endpoints to determine the behavior of the cubic function.

### Step 9: Evaluate the function at critical points
- \( f(-2) = (-2)^3 + 3(-2)^2 + 1 = -8 + 12 + 1 = 5 \)
- \( f(0) = 0^3 + 3(0)^2 + 1 = 1 \)

### Step 10: Determine the number of solutions
Since \( f(x) \) is a cubic polynomial and changes sign, it has one real root. Therefore, the cubic equation \( x^3 + 3x^2 + 1 = 0 \) has one real root.

### Conclusion
Adding the roots together, we have:
1. From \( x - 3 = 0 \), we have one solution \( x = 3 \).
2. From the cubic equation, we have one additional real root.

Thus, the total number of solutions for the equation \( x^4 - 6x^2 - 8x - 3 = 0 \) is:
\[ \text{Total solutions} = 1 + 1 = 2 \]

### Final Answer
The number of solutions is **2**.

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