Question

What is the product of the polynomials (x^2y+2)(x^2−y)?(1 point)
Responses

x3y−x2y2+2x2−2y
x cubed y minus x squared y squared plus 2 x squared minus 2 y

x4y−x2y2+2x2−2y
x superscript 4 baseline y minus x squared y squared plus 2 x squared minus 2 y

x4y+x2−2y
x superscript 4 baseline y plus x squared minus 2 y

x4y−xy+x2y

Answer 1

To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we can use the distributive property (also known as the FOIL method for binomials).

1. Distributing each term in the first polynomial to each term in the second polynomial:

\[
(x^2y)(x^2) + (x^2y)(-y) + (2)(x^2) + (2)(-y)
\]

2. Now, performing the multiplications:

- \(x^2y \cdot x^2 = x^4y\)
- \(x^2y \cdot (-y) = -x^2y^2\)
- \(2 \cdot x^2 = 2x^2\)
- \(2 \cdot (-y) = -2y\)

3. Now, putting it all together, we have:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]

Thus, the product of the polynomials \((x^2y + 2)(x^2 - y)\) is:

\[
\boxed{x^4y - x^2y^2 + 2x^2 - 2y}
\]

This corresponds to the second response, which states \(x^4y - x^2y^2 + 2x^2 - 2y\).