Find the continued product of `x+y,x-y,x^(2)+xy+y^(2),x^(2)-xy+y^(2)`

Answer

To find the continued product of the expressions \(x+y\), \(x-y\), \(x^2 + xy + y^2\), and \(x^2 - xy + y^2\), we will follow these steps:

### Step 1: Write down the expressions
We have the following expressions:
1. \(A = x + y\)
2. \(B = x - y\)
3. \(C = x^2 + xy + y^2\)
4. \(D = x^2 - xy + y^2\)

### Step 2: Multiply the first two expressions
We start by multiplying \(A\) and \(B\):
\[ A \cdot B = (x + y)(x - y) = x^2 - y^2 \]
This is based on the difference of squares formula.

### Step 3: Multiply the result with \(C\)
Next, we multiply the result \(x^2 - y^2\) with \(C\):
\[ (x^2 - y^2)(x^2 + xy + y^2) \]
Using the formula for the product of a sum and difference, we can expand this:
\[ = x^4 + x^3y + x^2y^2 - y^2x^2 - y^3x - y^4 \]
This simplifies to:
\[ = x^4 + x^3y - y^3x - y^4 \]

### Step 4: Multiply the result with \(D\)
Now we multiply the result with \(D\):
\[ (x^4 + x^3y - y^3x - y^4)(x^2 - xy + y^2) \]
This requires distributing each term in the first polynomial with each term in the second polynomial.

### Step 5: Simplify the expression
After distributing and combining like terms, we will find that:
\[ = x^6 - y^6 \]
This is based on the identity for the difference of cubes.

### Final Result
Thus, the continued product of the four given expressions is:
\[ x^6 - y^6 \]