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).

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

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

Now, combine all these results:

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

Rearranging the terms in standard polynomial form gives us:

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

So, filling in the blanks in your format:

- \(x^3y\): coefficient is 1 (so in the blank it’s just \(1\))
- \(-2x^2\): coefficient is -2
- \(xy\): coefficient is 1
- constant term: -2

Thus, the simplified expression is:

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

The final answer is:
\[
x^3y - 2x^2 + xy - 2
\]

So, the coefficients you need to fill in the blanks are:

\[
1, -2, 1, -2
\]