Question

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

Answer 1

To simplify the polynomial expression \((xy - 2)(x^2 + 1)(xy - 2)(x^2 + 1)\), we can first rewrite it for clarity:

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

Now, we'll first simplify the expression inside the parentheses, \((xy - 2)(x^2 + 1)\):

\[
(xy - 2)(x^2 + 1) = xy \cdot x^2 + xy \cdot 1 - 2 \cdot x^2 - 2 \cdot 1
\]
\[
= xy^3 + xy - 2x^2 - 2
\]

Next, we need to square this result:

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

To square this expression, we use the formula \((a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2(ab + ac + ad + bc + bd + cd)\).

Assign:
- \(a = xy^3\)
- \(b = xy\)
- \(c = -2x^2\)
- \(d = -2\)

Calculating \(a^2\), \(b^2\), \(c^2\), and \(d^2\):

- \(a^2 = (xy^3)^2 = x^2y^6\)
- \(b^2 = (xy)^2 = x^2y^2\)
- \(c^2 = (-2x^2)^2 = 4x^4\)
- \(d^2 = (-2)^2 = 4\)

Calculating the cross-products:

- \(ab = (xy^3)(xy) = x^2y^4\)
- \(ac = (xy^3)(-2x^2) = -2x^3y^3\)
- \(ad = (xy^3)(-2) = -2xy^3\)
- \(bc = (xy)(-2x^2) = -2x^3y\)
- \(bd = (xy)(-2) = -2xy\)
- \(cd = (-2x^2)(-2) = 4x^2\)

Now, we can combine everything:

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

Collecting like terms yields:

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

So, the final simplified expression is:

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

This polynomial presents the simplified version of the original expression.