Question

Simplify the polynomial expression (xy−2)(x2+1)(xy−2)(x2+1).(2 points) x3y+x3y+x2+x2+xy+xy+

Answer 1

To simplify the polynomial expression given, let's first rewrite it clearly. The expression appears to involve a product of two identical factors:

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

This can be simplified by recognizing that:

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

Next, we need to simplify the expression inside the parentheses:

1. **Expand \((xy - 2)(x^2 + 1)\)**:
\[
(xy - 2)(x^2 + 1) = xy \cdot x^2 + xy \cdot 1 - 2 \cdot x^2 - 2 \cdot 1
\]
\[
= x^3y + xy - 2x^2 - 2
\]

2. **Now square that result**:
\[
(x^3y + xy - 2x^2 - 2)^2
\]
To do this, we can use the formula \((a + b + c + d)^2\):
\[
= a^2 + b^2 + c^2 + d^2 + 2(ab + ac + ad + bc + bd + cd)
\]
where:
- \(a = x^3y\)
- \(b = xy\)
- \(c = -2x^2\)
- \(d = -2\)

**Calculating the squares**:
- \(a^2 = (x^3y)^2 = x^6y^2\)
- \(b^2 = (xy)^2 = x^2y^2\)
- \(c^2 = (-2x^2)^2 = 4x^4\)
- \(d^2 = (-2)^2 = 4\)

**Calculating the cross products**:
- \(ab = x^3y \cdot xy = x^4y^2\)
- \(ac = x^3y \cdot (-2x^2) = -2x^5y\)
- \(ad = x^3y \cdot (-2) = -2x^3y\)
- \(bc = xy \cdot (-2x^2) = -2x^3y\)
- \(bd = xy \cdot (-2) = -2xy\)
- \(cd = (-2x^2)(-2) = 4x^2\)

3. **Combine everything**:
After calculating all the terms, we group like terms:

\[
x^6y^2 + x^2y^2 + 4x^4 + 4 + 2(x^4y^2 - 2x^5y - 4x^3y - 2xy + 4x^2)
\]

This expression can be rearranged by collecting terms together:

\[
= x^6y^2 + x^2y^2 + 4x^4 + 4 + 2x^4y^2 - 4x^5y - 8x^3y - 4xy + 8x^2
\]

Finally, combining like terms will give us the fully simplified expression.

Alternatively, based on your question context, the second part you mentioned, **x^3y + x^3y + x^2 + x^2 + xy + xy**, simplifies quite easily:

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

The overall simplifications yield that the complete result would be a combination accordingly to combine with the coefficient scale.

Please confirm any specific paths or the exact elements you want further detailed or reanalyzed in polynomial simplification.