Question

What is the product of the polynomials (x^2y + 2) (x^2 - y)?
x^4y - x^2y^2 + 2x^2 - 2y
x^4y - xy + x^2y
x^4y = x^2 - 2y
x^3y - x^2y^2 + 2x^2 - 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). We will multiply each term in the first polynomial by each term in the second polynomial:

1. Multiply \(x^2y\) by \(x^2\):
\[
x^2y \cdot x^2 = x^4y
\]

2. Multiply \(x^2y\) by \(-y\):
\[
x^2y \cdot (-y) = -x^2y^2
\]

3. Multiply \(2\) by \(x^2\):
\[
2 \cdot x^2 = 2x^2
\]

4. Multiply \(2\) by \(-y\):
\[
2 \cdot (-y) = -2y
\]

Now we combine all these results together:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]

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