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

that is incorrect
blankx^3y+blankxy+blank

Answer 3

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 4

what about the second blank

Answer 5

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 6

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

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.