If `(x+i y)^3=u+i v ,`then Find p if `u/x+v/y=p(x^2-y^2)`.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to find \( p \) such that

\[ \frac{u}{x} + \frac{v}{y} = p(x^2 - y^2) \]

given that \( (x + i y)^3 = u + i v \).

### Step 1: Expand \( (x + i y)^3 \)

Using the binomial theorem, we can expand \( (x + i y)^3 \):

\[ (x + i y)^3 = x^3 + 3x^2(i y) + 3x(i y)^2 + (i y)^3 \]

Calculating each term:

- \( (i y)^2 = -y^2 \)
- \( (i y)^3 = -i y^3 \)

Thus, we have:

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

Combining the terms gives:

\[ = x^3 - 3xy^2 + i(3x^2y - y^3) \]

### Step 2: Identify \( u \) and \( v \)

From the expansion, we can identify:

\[ u = x^3 - 3xy^2 \]
\[ v = 3x^2y - y^3 \]

### Step 3: Substitute \( u \) and \( v \) into the left-hand side

Now, we substitute \( u \) and \( v \) into the expression \( \frac{u}{x} + \frac{v}{y} \):

\[ \frac{u}{x} + \frac{v}{y} = \frac{x^3 - 3xy^2}{x} + \frac{3x^2y - y^3}{y} \]

This simplifies to:

\[ = x^2 - 3y^2 + 3xy - y^2 \]

Combining like terms gives:

\[ = x^2 + 3xy - 4y^2 \]

### Step 4: Rearranging to find \( p \)

We want to express this in the form \( p(x^2 - y^2) \). Notice that:

\[ x^2 + 3xy - 4y^2 = 4(x^2 - y^2) \]

This means:

\[ p = 4 \]

### Final Answer

Thus, the value of \( p \) is:

\[ \boxed{4} \]