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

Question

Responses

A) x^3y−x^2y^2+2x^2−2y
x cubed y minus x squared y squared plus 2 x squared minus 2 y

B) x^4y−x^2y^2+2x^2−2y
x superscript 4 baseline y minus x squared y squared plus 2 x squared minus 2 y

C) x^4y+x^2−2y
x superscript 4 baseline y plus x squared minus 2 y

D) x^4y−xy+x^2y
x superscript 4 baseline y minus x y plus x squared y

Answer 1

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

We can expand the expression step-by-step:

1. Distribute \(x^2y\):

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

2. Distribute \(2\):

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

Now combine all the terms we obtained: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

Thus, the product of the polynomials \((x^2y + 2)(x^2 - y)\) is: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

Looking at the provided options, the expression matches with:

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

So, the correct answer is B.