Question

simplify the polynomial expression then fill in the blanks (_)
(xy - 2)(x^2 + 1)

_x^3y + _ x^2 + _xy + _

Answer 1

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

Let's distribute \(xy - 2\) over \(x^2 + 1\):

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

2. Multiply \(xy\) by \(1\):
\[
xy \cdot 1 = xy
\]

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

4. Multiply \(-2\) by \(1\):
\[
-2 \cdot 1 = -2
\]

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

Rearranging this, we have:
\[
x^3y - 2x^2 + xy - 2
\]

Now, we can fill in the blanks:
\[
_x^3y + _ x^2 + _xy + _
\]
This corresponds to:
- 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**.

Filling in the blanks:
\[
1, -2, 1, -2
\]

Thus, the filled-in blanks are:
\[
(1)(-2)(1)(-2)
\]