Question

Simplify the polynomial expression (xy - 2) (x^2 + 1).
___x^3y + ___x^2 + ___xy + ____

Answer 1

To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will use the distributive property (also known as the FOIL method for binomials). Here's how to do it:

1. Distribute \(xy\) to both terms in \( (x^2 + 1) \):
\[
xy \cdot x^2 = x^3y
\]
\[
xy \cdot 1 = xy
\]

2. Distribute \(-2\) to both terms in \( (x^2 + 1) \):
\[
-2 \cdot x^2 = -2x^2
\]
\[
-2 \cdot 1 = -2
\]

Now we combine all the results:
\[
x^3y + xy - 2x^2 - 2
\]

Next, we can rearrange this expression into standard form, typically ordered by decreasing powers of \(x\):
\[
x^3y - 2x^2 + xy - 2
\]

Finally, we fill in the blanks for your formatted answer:
- Coefficient of \(x^3y\): \(1\)
- Coefficient of \(x^2\): \(-2\)
- Coefficient of \(xy\): \(1\)
- Constant term: \(-2\)

So, the final answer is:
___\(1\)___ \(x^3y\) + ___\(-2\)___ \(x^2\) + ___\(1\)___ \(xy\) + ___\(-2\)___