What does it mean for a set to be "closed"? (1 point)

The correct definition for a "closed" set in the context of mathematics is:

**An operation performed on members of that set will result in a member of the same set.**

This means that if you take any two elements from the set and apply a specific operation (like addition, multiplication, etc.), the result will also be an element of that set. For example, the set of integers is closed under addition because the sum of any two integers is also an integer.

Add the polynomial expressions 2r^2 - 3r + 4 and - 2r^2 + 3r + 6

To add the polynomial expressions \(2r^2 - 3r + 4\) and \(-2r^2 + 3r + 6\), we can group and combine like terms.

1. Start with the expressions:
\[
2r^2 - 3r + 4
\]
\[
-2r^2 + 3r + 6
\]

2. Add the corresponding coefficients:
- For \(r^2\) terms: \(2r^2 + (-2r^2) = 0r^2\)
- For \(r\) terms: \(-3r + 3r = 0r\)
- For constant terms: \(4 + 6 = 10\)

3. Combine the results:
\[
0r^2 + 0r + 10 = 10
\]

Thus, the result of adding the two polynomial expressions is:
\[
\boxed{10}
\]

Add the expressions.

Expression 1: -3k + 4k^3 - 2
Expression 2: 9k^2 - 3k^3 + 4k - 3

To add the expressions \(-3k + 4k^3 - 2\) and \(9k^2 - 3k^3 + 4k - 3\), we will combine like terms.

**Step 1: Write down the expressions.**

Expression 1:
\[
-3k + 4k^3 - 2
\]

Expression 2:
\[
9k^2 - 3k^3 + 4k - 3
\]

**Step 2: Rearrange and combine like terms.**

We will arrange the terms by their degree:

\[
(4k^3 - 3k^3) + 9k^2 + (-3k + 4k) + (-2 - 3)
\]

**Step 3: Combine the coefficients for each type of term.**

- For \(k^3\) terms:
\[
4k^3 - 3k^3 = 1k^3
\]

- For \(k^2\) terms:
\[
9k^2 \quad \text{(since there is no corresponding } k^2 \text{ in Expression 1)}
\]

- For \(k\) terms:
\[
-3k + 4k = 1k
\]

- For the constant terms:
\[
-2 - 3 = -5
\]

**Step 4: Write the result.**

Combining everything, we have:

\[
1k^3 + 9k^2 + 1k - 5
\]

This can be simplified as:

\[
k^3 + 9k^2 + k - 5
\]

Thus, the result of adding the two expressions is:
\[
\boxed{k^3 + 9k^2 + k - 5}
\]

this is the equation

?k^3 + ?k² + ?k + ?

To express the result in the format of the equation \( ?k^3 + ?k^2 + ?k + ? \) with the coefficients filled in, we can write:

From the previous addition we calculated:
\[
k^3 + 9k^2 + k - 5
\]

The coefficients are as follows:

- Coefficient of \(k^3\): **1**
- Coefficient of \(k^2\): **9**
- Coefficient of \(k\): **1**
- Constant term: **-5**

Thus, the equation can be expressed as:
\[
1k^3 + 9k^2 + 1k - 5
\]

So the coefficients in the format \( ?k^3 + ?k^2 + ?k + ? \) are:

- \(1\) for \(k^3\)
- \(9\) for \(k^2\)
- \(1\) for \(k\)
- \(-5\) for the constant

The filled-in equation is:
\[
1k^3 + 9k^2 + 1k - 5
\]

1k^3 + 9k^2 + 1k + ?

From the previous addition, we found that the result of adding the two polynomial expressions was:

\[
k^3 + 9k^2 + k - 5
\]

To fit this into the format \(1k^3 + 9k^2 + 1k + ?\), we identify the constant term.

The constant term in the equation is \(-5\).

Thus, filling in the last part of your equation gives us:

\[
1k^3 + 9k^2 + 1k - 5
\]

So, the final answer is:

\[
1k^3 + 9k^2 + 1k + (-5)
\]

If you're looking to fill in the blank:

The filled-in expression is:

\[
1k^3 + 9k^2 + 1k - 5
\]

1k^3 + 9k^2 + 1k + ?

its not 5

I apologize for any confusion earlier. Based on your equation format, let's clarify:

From our addition of the polynomials, we arrived at:

\[
k^3 + 9k^2 + k - 5
\]

In this expression:

- The coefficient of \(k^3\) is **1**.
- The coefficient of \(k^2\) is **9**.
- The coefficient of \(k\) is **1**.
- The constant term is **-5**.

Therefore, if we represent it in your format \( 1k^3 + 9k^2 + 1k + ? \), the appropriate value to fill in the last part is:

\[
-5
\]

So, the full equation should read:

\[
1k^3 + 9k^2 + 1k - 5
\]

If you are looking for a different value or there is a misunderstanding regarding which constant to include, please provide clarification. Thank you!

This is the equation

1k^3 + 9k^2 + 1k + (blank)

Based on the addition of the expressions provided earlier, the complete result we found was:

\[
k^3 + 9k^2 + k - 5
\]

In the format you provided, which is:

\[
1k^3 + 9k^2 + 1k + (blank)
\]

The blank should be filled with the constant term from the polynomial result.

In this case, the constant term is **-5**. Therefore, filling in the blank gives you:

\[
1k^3 + 9k^2 + 1k + (-5)
\]

If you're asking for just the constant term, it is:

\(-5\).

So the complete expression, including the blank, is:

\[
1k^3 + 9k^2 + 1k + (-5)
\]

If you're looking for a different aspect or a specific value, please let me know!

Simplify (9x^2 - 6x + 2) - (x^2 + 4x - 39)

The difference of the polynomials is (blank)

To simplify the expression \((9x^2 - 6x + 2) - (x^2 + 4x - 39)\), we will first distribute the negative sign across the second polynomial, and then combine the like terms.

**Step 1: Distribute the negative sign**

\[
(9x^2 - 6x + 2) - (x^2 + 4x - 39) = 9x^2 - 6x + 2 - x^2 - 4x + 39
\]

**Step 2: Combine like terms**

- For \(x^2\) terms:
\[
9x^2 - x^2 = 8x^2
\]

- For \(x\) terms:
\[
-6x - 4x = -10x
\]

- For constant terms:
\[
2 + 39 = 41
\]

**Step 3: Write the final expression**

Combining all of the results, we have:
\[
8x^2 - 10x + 41
\]

So, the simplified difference of the polynomials is:

\[
\boxed{8x^2 - 10x + 41}
\]

Fill in the blanks to complete the polynomial equation that shows the subtraction of the second expression from the first and the resulting difference. (2 points)

9x^2 + ?x + 13
2x^2 + 12x + ?
?x^2 + 5x + 2

To fill in the blanks in the polynomial equation that shows the subtraction of the second expression from the first, we will use the two polynomial expressions provided. The operation will be:

**First Expression:**
\[
9x^2 + ?x + 13
\]

**Second Expression:**
\[
2x^2 + 12x + ?
\]

We will perform the subtraction:

\[
(9x^2 + ?x + 13) - (2x^2 + 12x + ?)
\]

To fill in the blanks, we first need to calculate the valid coefficients that will lead us to the resulting polynomial shown below.

**Resulting Polynomial:**
\[
?x^2 + 5x + 2
\]

### Step 1: Set Up the Subtraction

1. We subtract the \(x^2\) terms:
\[
9x^2 - 2x^2 = ?x^2 \quad \Rightarrow \quad 9 - 2 = 7 \quad \Rightarrow \quad ? = 7
\]

2. Next, subtract the \(x\) terms:
\[
?x - 12x = 5x \quad \Rightarrow \quad ? - 12 = 5 \quad \Rightarrow \quad ? = 5 + 12 \quad \Rightarrow \quad ? = 17
\]

3. Finally, subtract the constant terms:
\[
13 - ? = 2 \quad \Rightarrow \quad 13 - ? = 2 \quad \Rightarrow \quad ? = 13 - 2 \quad \Rightarrow \quad ? = 11
\]

### Final Fill-in Summary:
Now we can fill in the blanks:

- For the first expression \(9x^2 + 17x + 13\)
- For the second expression \(2x^2 + 12x + 11\)
- The resulting expression is \(7x^2 + 5x + 2\)

So, the complete polynomial equations become:

1. **First Expression:** \(9x^2 + 17x + 13\)
2. **Second Expression:** \(2x^2 + 12x + 11\)
3. **Resulting Difference:** \(7x^2 + 5x + 2\)

Thus, the final blame is:

**Filled in Answer:**
- \(17\) for the first blank (coefficient of \(x\))
- \(11\) for the second blank (constant term)
- \(7\) for the third blank (coefficient of \(x^2\))

The completed result is:
\[
\begin{align*}
9x^2 + 17x + 13 \\
2x^2 + 12x + 11 \\
7x^2 + 5x + 2
\end{align*}
\]

Simplify (1/4x^2 - 3x + 2.25) + (0.75x^2 + 2x - 1/4)

To simplify the expression \((\frac{1}{4}x^2 - 3x + 2.25) + (0.75x^2 + 2x - \frac{1}{4})\), we will first combine like terms.

### Step 1: Rewrite the expression
We can express the whole expression as follows:

\[
\left(\frac{1}{4}x^2 - 3x + 2.25\right) + \left(0.75x^2 + 2x - \frac{1}{4}\right)
\]

### Step 2: Combine coefficients for like terms
**For \(x^2\) terms:**

\[
\frac{1}{4}x^2 + 0.75x^2
\]

We can convert \(0.75\) into a fraction:

\[
0.75 = \frac{3}{4}
\]

Now, add the \(x^2\) terms:

\[
\frac{1}{4}x^2 + \frac{3}{4}x^2 = \frac{1 + 3}{4}x^2 = \frac{4}{4}x^2 = 1x^2
\]

**For \(x\) terms:**

\[
-3x + 2x = -3x + 2x = -1x
\]

**For constant terms:**

Convert \(2.25\) into a fraction:

\[
2.25 = \frac{9}{4}
\]

Now combine the constants:

\[
\frac{9}{4} - \frac{1}{4} = \frac{9 - 1}{4} = \frac{8}{4} = 2
\]

### Step 3: Write the result
Now combining all these results:

\[
1x^2 - 1x + 2 = x^2 - x + 2
\]

So the simplified expression is:

\[
\boxed{x^2 - x + 2}
\]