The equation `x^(3)+y^(3)=0` represents

Options

A. three real straight lines
B. three points
C. the combined euation of a straight line and a circle
D. none of these

Explanation

To solve the equation \( x^3 + y^3 = 0 \) and determine what it represents, we can follow these steps:

### Step 1: Factor the equation
The equation \( x^3 + y^3 = 0 \) can be factored using the sum of cubes formula:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) = 0 \]

### Step 2: Set each factor to zero
From the factored form, we can set each factor equal to zero:
1. \( x + y = 0 \)
2. \( x^2 - xy + y^2 = 0 \)

### Step 3: Analyze the first factor
The first equation \( x + y = 0 \) represents a straight line in the Cartesian plane. This line has a slope of -1 and passes through the origin.

### Step 4: Analyze the second factor
Now, we look at the second factor \( x^2 - xy + y^2 = 0 \). This is a quadratic equation in terms of \( x \) and \( y \). To find the roots, we can use the discriminant method:
\[ D = b^2 - 4ac = (-y)^2 - 4(1)(y^2) = y^2 - 4y^2 = -3y^2 \]
Since the discriminant \( D \) is less than zero (for \( y \neq 0 \)), this means that the quadratic has no real solutions, but it has two complex (imaginary) solutions.

### Step 5: Conclusion
Thus, the equation \( x^3 + y^3 = 0 \) represents:
- One real straight line from \( x + y = 0 \)
- Two imaginary straight lines from \( x^2 - xy + y^2 = 0 \)

### Final Answer
The equation \( x^3 + y^3 = 0 \) represents one real straight line and two imaginary straight lines.

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