What is the product of the polynomials (x2y+2)(x2−y) ?(1 point)

Question

Responses

x4y−xy+x2y
x superscript 4 baseline y minus x y plus x squared y

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

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

x4y−x2y2+2x2−2y

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). Let's break it down step-by-step:

1. Distribute each term in the first polynomial by each term in the second polynomial:

   • \(x^2y \cdot x^2\) gives \(x^4y\)
   • \(x^2y \cdot (-y)\) gives \(-x^2y^2\)
   • \(2 \cdot x^2\) gives \(2x^2\)
   • \(2 \cdot (-y)\) gives \(-2y\)

2. Combine all the terms: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

Thus, the final product is \(x^4y - x^2y^2 + 2x^2 - 2y\).

Now let's match this with the provided options:

• \(x^4y - x^2y^2 + 2x^2 - 2y\)

This matches the fourth response option: x4y−x2y2+2x2−2y.

So, the correct answer is:

x4y−x2y2+2x2−2y.