Question

What is the product of the polynomials $(x^{2}y+2)(x^{2}-y)$ ? (1 point)
$x^{4}y-xy+x^{2}y$
$x^{3}y-x^{2}y^{2}+2x^{2}-2y$
$x^{4}y+x^{2}-2y$
$x^{4}y-x^{2}y^{2}+2x^{2}-2y$

Answer 1

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

1. Distributing \(x^{2}y\) over \(x^{2} - y\):
\[
x^{2}y \cdot x^{2} = x^{4}y
\]
\[
x^{2}y \cdot (-y) = -x^{2}y^{2}
\]
So from \(x^{2}y\), we get:
\[
x^{4}y - x^{2}y^{2}
\]

2. Distributing \(2\) over \(x^{2} - y\):
\[
2 \cdot x^{2} = 2x^{2}
\]
\[
2 \cdot (-y) = -2y
\]
So from \(2\), we get:
\[
2x^{2} - 2y
\]

Now, putting it all together, we add all the terms from the distributions:
\[
x^{4}y - x^{2}y^{2} + 2x^{2} - 2y
\]

Thus, the final result of the product is:
\[
\boxed{x^{4}y - x^{2}y^{2} + 2x^{2} - 2y}
\]

This matches with the corresponding option given in the question:
\(x^{4}y - x^{2}y^{2} + 2x^{2} - 2y\).