Question

simplify the polynomial expression then fill in the blanks (_)
(y^2 - 3) (xy + 1)
_xy^3 + _ xy + _ y^2 + _

Answer 1

To simplify the polynomial expression \((y^2 - 3)(xy + 1)\), we'll distribute each term in the first polynomial by each term in the second polynomial.

1. Distributing \(y^2\):
\[
y^2 \cdot (xy) = xy^3
\]
\[
y^2 \cdot 1 = y^2
\]

2. Distributing \(-3\):
\[
-3 \cdot (xy) = -3xy
\]
\[
-3 \cdot 1 = -3
\]

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

Let's rearrange it in standard form:
\[
xy^3 - 3xy + y^2 - 3
\]

Now we fill in the blanks in the provided format:
- The coefficient of \(y^3\) (which is \(xy^3\)) corresponds to \(1\) (there's no coefficient shown).
- The coefficient of \(xy\) (which is \(-3xy\)) corresponds to \(-3\).
- The coefficient of \(y^2\) corresponds to \(1\) (as \(y^2\)).
- The constant term corresponds to \(-3\).

Thus, the final values to fill in the blanks are:

1. Coefficient of \(y^3\): **1**
2. Coefficient of \(xy\): **-3**
3. Coefficient of \(y^2\): **1**
4. Constant: **-3**

So, the result is:
\[
(1, -3, 1, -3)
\]