Question

Simplify the polynomial expression (xy−2)(x2+1) .(2 points) x3y+ x2+ xy+

Answer 1

that is incorrect
blankx^3y+blankxy+blank

Answer 2

what about the second blank

Answer 3

blankx^3y+blankx^2+blankxy+blank

Answer 4

To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we can use the distributive property (also known as the FOIL method in the case of binomials). Here's how we do it step-by-step:

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

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

3. Now we combine all the terms from steps 1 and 2:
\[
x^3y + xy - 2x^2 - 2
\]

4. Rearranging the terms gives us:
\[
x^3y - 2x^2 + xy - 2
\]

This is the simplified form of the polynomial expression. Therefore, the answer is:
\[
x^3y + xy - 2x^2 - 2
\]

Answer 5

Apologies for the misunderstanding. Let's go through the simplification of the expression \((xy - 2)(x^2 + 1)\) step-by-step to match the format you requested:

1. **Distribute each term in \(xy - 2\) to each term in \(x^2 + 1\)**:

- **First, distribute \(xy\)**:
\[
xy \cdot x^2 = x^3y
\]
\[
xy \cdot 1 = xy
\]

- **Next, distribute \(-2\)**:
\[
-2 \cdot x^2 = -2x^2
\]
\[
-2 \cdot 1 = -2
\]

2. **Combine all the results**:
\[
(xy - 2)(x^2 + 1) = x^3y + xy - 2x^2 - 2
\]

Now, let's express the result in the form you provided:

- **Coefficient of \(x^3y\)**: 1
- **Coefficient of \(xy\)**: 1
- **Constant term (or constant part)**: -2

So, putting that into the blank format you asked for, we have:

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

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

So the simplified polynomial expression in the form you wanted is:
- **1** (for \(x^3y\))
- **1** (for \(xy\))
- **−2** (constant term)

Again, I apologize for the previous oversight! Thank you for your patience.

Answer 6

I see now that you're specifically asking for the blanks in the expression \(x^3y + xy + \_\_\_\_\_\_\_\_\_\_\_\_\_\). Let’s clarify the final result for the expression \((xy - 2)(x^2 + 1)\).

After expanding the expression, we found:

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

If you're looking for the structure:

\[
x^3y + xy + \_\_\_\_\_\_\_\_\_\_\_\_\_\,
\]

the terms in the resulting polynomial are:

1. Coefficient of \(x^3y\): **1**
2. Coefficient of \(xy\): **1**
3. The remaining part (constant term or what's left after those): **-2x^2 - 2**.

So you can fill in the blanks this way:

- For the first blank (associated with \(x^3y\)): **1**
- For the second blank (associated with \(xy\)): **1**
- For the remaining part (the constant term or \(x^0\) term): **-2x^2 - 2**

In total, if you're looking for the contributions for just the second blank (the constant term), based on the original expansion it's -2.

So, to summarize:

**Final Answer in the format you requested**:

- For \(x^3y\): **1**
- For \(xy\): **1**
- For the constant term: **-2**

If you meant for a second blank specifically based on the terms, please let me know!

Answer 7

Thank you for your clarification! Based on the expanded polynomial expression from \((xy - 2)(x^2 + 1)\), let's rewrite it in the requested format with specific coefficients for each term:

We derived:
\[
x^3y + xy - 2x^2 - 2.
\]
To put this in the form:
\[
\text{(coefficient)}x^3y + \text{(coefficient)}x^2 + \text{(coefficient)}xy + \text{(constant term)},
\]
we can identify the coefficients for each term as follows:

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

So, filling in each blank:

\[
\text{1} \cdot x^3y + \text{-2} \cdot x^2 + \text{1} \cdot xy + \text{-2}
\]

Thus, the final expression in the requested format would be:

**Final answer:**
- **1** \(x^3y\) +
- **-2** \(x^2\) +
- **1** \(xy\) +
- **-2**

I hope this helps! Thank you for your patience.