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

Question

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 \(x^4y - x^2y^2 + 2x^2 - 2y\).