Question

Simplify the polynomial expression (xy−2)(x^2+1)
.(2 points)
_x^3 y+_x2+_xy+_

Answer 1

To simplify the expression \((xy - 2)(x^2 + 1)\), we will use the distributive property (also known as the FOIL method for binomials).

1. Distribute \(xy\) across \(x^2 + 1\):
\[
xy \cdot x^2 + xy \cdot 1 = xyx^2 + xy = x^3y + xy
\]

2. Distribute \(-2\) across \(x^2 + 1\):
\[
-2 \cdot x^2 + -2 \cdot 1 = -2x^2 - 2
\]

3. Now, combine all the terms:
\[
x^3y + xy - 2x^2 - 2
\]

Putting it all together, the simplified form of the polynomial expression is:

\[
x^3y - 2x^2 + xy - 2
\]

Therefore, filling in the blanks for your polynomial expression:
- The coefficient for \(x^3y\) is **1**.
- The coefficient for \(x^2\) is **-2**.
- The coefficient for \(xy\) is **1**.
- The constant term is **-2**.

So the final answer is:
\[
1 \, x^3 y - 2 \, x^2 + 1 \, xy - 2
\]