Question

Simplify the polynomial expression (xy−2)(x2+1) .(2 points) []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).

\[
(xy - 2)(x^2 + 1)
\]

Distributing each term in the first polynomial to each term in the second polynomial:

1. \(xy \cdot x^2 = x^3y\)
2. \(xy \cdot 1 = xy\)
3. \(-2 \cdot x^2 = -2x^2\)
4. \(-2 \cdot 1 = -2\)

Now, we can combine all these results:

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

Now, we can write it in descending order by the power of \(x\):

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

Thus, the simplified expression is:

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

In the form you provided, it can be filled in as:

- Coefficient of \(x^3y\) is \(1\)
- Coefficient of \(x^2\) is \(-2\)
- Coefficient of \(xy\) is \(1\)
- Constant term is \(-2\)

So the final representation is:

\[
[x^3y] + [-2]x^2 + [1]xy + [-2]
\]