Consider the expression z+6/3^2 - (2 x 3) + 5 What is the simplified denominator?

Let's simplify the expression step by step:

The original expression is:

\[ z + \frac{6}{3^2} - (2 \times 3) + 5 \]

1. Calculate \( 3^2 \):
\[ 3^2 = 9 \]

2. Now substituting it back in:
\[ z + \frac{6}{9} - (2 \times 3) + 5 \]

3. Simplify \( \frac{6}{9} \) (which reduces to \( \frac{2}{3} \)):
\[ z + \frac{2}{3} - (2 \times 3) + 5 \]

4. Calculate \( 2 \times 3 \):
\[ 2 \times 3 = 6 \]

5. Now substituting it back:
\[ z + \frac{2}{3} - 6 + 5 \]

6. Combine \( -6 + 5 \):
\[ -6 + 5 = -1 \]

7. Now substituting back:
\[ z + \frac{2}{3} - 1 \]

8. Rewrite \( -1 \) as \( -\frac{3}{3} \):
\[ z + \frac{2}{3} - \frac{3}{3} \]

9. Combine the fractions:
\[ z + \left(\frac{2}{3} - \frac{3}{3}\right) = z - \frac{1}{3} \]

Thus, there is no denominator for the entire expression after simplification; however, if you're asking for the denominator from the fraction \( \frac{2}{3} \) that appeared in the calculation, it's 3.

The answer choices (8 and 26) do not relate to the simplified expression \( z - \frac{1}{3} \). Could you please confirm the question, as it seems your provided options don’t match the denominator present in the calculations? If you're looking for something else, please clarify!

choose one of the options

Given the options A. 8 and B. 26, and the context of the calculation, let's look at the simplified expression extracted from the original problem, where the fraction we calculated was \( \frac{2}{3} \).

However, neither option 8 nor 26 is relevant to any denominator in the simplification itself. If I must choose one based solely on available options without more context, I'll arbitrarily choose:

**A. 8**

Please let me know if there's more information that can help refine the response!

Which of the following correctly demonstrates the Commutative Property of Addition?

5z^9 + 8a = 8a + 5z^9
5 + 8 = 4 + 9

The Commutative Property of Addition states that changing the order of the addends does not change the sum. In other words, for any numbers \( a \) and \( b \), the following holds:

\[ a + b = b + a \]

Now, let's examine the provided options:

1. **\( 5z^9 + 8a = 8a + 5z^9 \)**: This shows two terms being added in different orders, which illustrates the Commutative Property of Addition. It states that adding \( 5z^9 \) and \( 8a \) yields the same result as adding \( 8a \) and \( 5z^9 \).

2. **\( 5 + 8 = 4 + 9 \)**: This expression does not demonstrate the Commutative Property of Addition; instead, it is simply an equation that is equal on both sides but does not show a rearranging of addends.

Therefore, the correct option that demonstrates the Commutative Property of Addition is:

**\( 5z^9 + 8a = 8a + 5z^9 \)**.

Which expression is equivalent to (3r * 5) * (- 3 * 8)'

-45r * 5
24r(5 * -3)

To find an expression equivalent to \((3r * 5) * (-3 * 8)\), let's first simplify the expression step by step.

1. Simplify \(-3 * 8\):
\[
-3 * 8 = -24
\]

2. Substitute \(-24\) back into the expression:
\[
(3r * 5) * (-24)
\]

3. Now calculate \(3r * 5\):
\[
3r * 5 = 15r
\]

4. Substitute this back into the expression:
\[
15r * (-24) = -360r
\]

Now, let's analyze the given options:

1. **\(-45r * 5\)**: Simplifying this results in:
\[
-45r * 5 = -225r
\]
This is not equivalent to \(-360r\).

2. **\(24r(5 * -3)\)**: Let's simplify this:
\[
5 * -3 = -15
\]
Therefore, it becomes:
\[
24r * -15 = -360r
\]
This is equivalent to our calculated expression.

Thus, the correct expression that is equivalent to \((3r * 5) * (-3 * 8)\) is:

**\(24r(5 * -3)\)**.

Use the Distributive Property to write the expression 4x^2 36 a different way.

(2x + 6)(2x - 6)
(x + 18)(x - 18)

To rewrite the expression \( 4x^2 - 36 \) using the Distributive Property, we can factor it.

The expression \( 4x^2 - 36 \) is a difference of squares, which can be factored using the formula:

\[
a^2 - b^2 = (a - b)(a + b)
\]

In this case:

- \( a^2 = 4x^2 \) implies \( a = 2x \)
- \( b^2 = 36 \) implies \( b = 6 \)

Hence, we can factor \( 4x^2 - 36 \) as:

\[
4x^2 - 36 = (2x - 6)(2x + 6)
\]

Now, let's examine the provided options:

1. **\((2x + 6)(2x - 6)\)**: This is the correct factored form of \( 4x^2 - 36 \) as shown above (noting the order of terms doesn't affect the product because of the commutative property).

2. **\((x + 18)(x - 18)\)**: This does not relate to our original expression.

Thus, the correct option using the Distributive Property to rewrite \( 4x^2 - 36 \) is:

**\((2x + 6)(2x - 6)\)**.

Find the expression that is equivalent to 3x^2 + 2x - 5x - 2x^2 + 2

(3x + 2)(x + 1)
(x - 2)(x - 1)

To find the expression equivalent to \(3x^2 + 2x - 5x - 2x^2 + 2\), let's first simplify the given expression step by step.

1. Combine like terms in the expression:
\[
3x^2 - 2x^2 + 2x - 5x + 2
\]

- Combine \(3x^2\) and \(-2x^2\):
\[
3x^2 - 2x^2 = x^2
\]

- Combine \(2x\) and \(-5x\):
\[
2x - 5x = -3x
\]

Therefore, the expression simplifies to:
\[
x^2 - 3x + 2
\]

2. Now, we need to factor the expression \(x^2 - 3x + 2\). We look for two numbers that multiply to \(2\) (the constant term) and add up to \(-3\) (the coefficient of \(x\)).

The numbers \(-1\) and \(-2\) meet these conditions:
\(-1 \times -2 = 2\)
and
\(-1 + (-2) = -3\).

3. Thus, we can factor the expression:
\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]

Now, let's look at the options provided:

1. **\((3x + 2)(x + 1)\)**: This does not relate to our simplified expression.

2. **\((x - 2)(x - 1)\)**: This matches the factored form we derived.

Therefore, the expression equivalent to \(3x^2 + 2x - 5x - 2x^2 + 2\) is:

\(\boxed{(x - 2)(x - 1)}\).

Which of the following shows that polynomials are a closed system under addition?

A. (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 1.846x^2 - 12.84x + 5.96
B. (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 10.994x^2 + 5.96

To determine which option shows that polynomials are a closed system under addition, we need to understand what it means for a system to be closed under addition. A set of numbers (or polynomials, in this case) is closed under addition if the sum of two elements from the set is also an element of the same set.

In the context of polynomials, it means that when you add two polynomials together, the result should also be a polynomial.

Now, let's analyze the provided options:

### Option A:
\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 1.846x^2 - 12.84x + 5.96
\]

1. **Adding the polynomials on the left:**
\[
0.923x^2 + 0.923x^2 = 1.846x^2
\]
\[
-6.42x - 6.42x = -12.84x
\]
\[
2.98 + 2.98 = 5.96
\]

So, it is correctly simplified to the right side:
\[
1.846x^2 - 12.84x + 5.96
\]

### Option B:
\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 10.994x^2 + 5.96
\]

1. **Adding the polynomials on the left:**
As seen before:
\[
0.923x^2 + 0.923x^2 = 1.846x^2
\]
\[
-6.42x - 6.42x = -12.84x
\]
\[
2.98 + 2.98 = 5.96
\]

The left-hand side was simplified to:
\[
1.846x^2 - 12.84x + 5.96
\]

However, the right side in this option is \(10.994x^2 + 5.96\), which is incorrect and does not align with the left side.

### Conclusion:
**Option correctly shows that adding two polynomials results in another polynomial, while **Option B** contains an incorrect statement. Therefore, Option A illustrates that polynomials form a closed system under addition.

Thus, the answer is:

**A. (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 1.846x^2 - 12.84x + 5.96**.

What must be true about this multiplication problem? (9x^3 + 7/6) x (x - 2/5) (1 point)

The result is a polynomial.
The result is not a polynomial.

To determine whether the result of the multiplication problem \((9x^3 + \frac{7}{6}) \times (x - \frac{2}{5})\) is a polynomial or not, we first need to understand what constitutes a polynomial.

**Definition of a Polynomial:**
A polynomial is a mathematical expression that can consist of terms, where each term is made up of a variable raised to a non-negative integer power, multiplied by a coefficient (which can be a real or complex number).

A polynomial can include multiple terms (such as \(ax^n + bx^{n-1} + ... + k\)), with each term having a coefficient and a variable raised to a non-negative integer exponent.

### Evaluating the Expression:

1. **First Expression: \(9x^3 + \frac{7}{6}\)**:
- This is a polynomial because it consists of a term \(9x^3\) (where the variable \(x\) is raised to a non-negative integer power) and a constant term \(\frac{7}{6}\).

2. **Second Expression: \(x - \frac{2}{5}\)**:
- This is also a polynomial because it contains \(x\) (which is \(x^1\)) and \(-\frac{2}{5}\) (a constant term).

### Multiplying the Two Polynomials:
When we multiply two polynomials, the result is always a polynomial. Let's look at the nature of the multiplication:

\[
(9x^3 + \frac{7}{6}) \times (x - \frac{2}{5})
\]

The resulting expression will be formed by distributing each term in the first polynomial to each term in the second polynomial:

- Multiplying \(9x^3\) by both terms in \((x - \frac{2}{5})\)
- Multiplying \(\frac{7}{6}\) by both terms in \((x - \frac{2}{5})\)

The result will be:

1. \(9x^3 \cdot x = 9x^4\)
2. \(9x^3 \cdot (-\frac{2}{5}) = -\frac{18}{5}x^3\)
3. \(\frac{7}{6} \cdot x = \frac{7}{6}x\)
4. \(\frac{7}{6} \cdot (-\frac{2}{5}) = -\frac{14}{30} = -\frac{7}{15}\)

### Conclusion:
The overall result will be:

\[
9x^4 - \frac{18}{5}x^3 + \frac{7}{6}x - \frac{7}{15}
\]

Since all parts of the final product involve non-negative integer exponents and real (or rational) coefficients, **the result is indeed a polynomial**.

Therefore, the correct answer is:

**The result is a polynomial.**

Which of the following is the simplified form of 5x^2 - 40x + 10/5 ?

x^2 - 35x + 5
x^2 - 8x + 2

To simplify the expression \( 5x^2 - 40x + \frac{10}{5} \), we will first simplify \(\frac{10}{5}\) and then factor the entire expression.

1. **Simplify \( \frac{10}{5} \)**:
\[
\frac{10}{5} = 2
\]

2. **Now substitute this back into the expression**:
\[
5x^2 - 40x + 2
\]

3. **Now, we can factor out the common factor of 5 from the first two terms**:
\[
5(x^2 - 8x) + 2
\]

4. **So far, we have**:
\[
5(x^2 - 8x + \frac{2}{5})
\]

However, since we are simplifying, let’s divide the entire expression \(5x^2 - 40x + 2\) by 5:

\[
5x^2 - 40x + 2 = 5 \left( x^2 - 8x + \frac{2}{5} \right)
\]

5. **The simplified form of the expression is**:
\[
x^2 - 8x + 2
\]

Now let’s check the options provided:

1. **\( x^2 - 35x + 5 \)**: This does not match.
2. **\( x^2 - 8x + 2 \)**: This is the correct form.

Thus, the answer is:

**\( x^2 - 8x + 2 \)**.

Which of the following correctly divides the expressions -21x^3 + 14x^2 - 14x/7x

3x^2 + 2x - 2
-3x^2 + 2x - 2

To divide the expression \(-21x^3 + 14x^2 - 14x\) by \(7x\), we will perform the division term by term.

### Step 1: Divide each term in the polynomial by \(7x\):

1. **First term: \(-21x^3 \div 7x\)**:
\[
-21x^3 \div 7x = -3x^2
\]

2. **Second term: \(14x^2 \div 7x\)**:
\[
14x^2 \div 7x = 2x
\]

3. **Third term: \(-14x \div 7x\)**:
\[
-14x \div 7x = -2
\]

### Step 2: Combine the results:
Putting everything together, we get:
\[
-3x^2 + 2x - 2
\]

### Conclusion:
Comparing this result with the options provided:

1. **\(3x^2 + 2x - 2\)**: This does not match.
2. **\(-3x^2 + 2x - 2\)**: This is the correct result.

Therefore, the correct answer is:

**\(-3x^2 + 2x - 2\)**.

What is the quotient of the following? 28x^7 + 40x^6 - 16x^4/4x^3

?x^? + ?x^? -?x

To find the quotient of the expression

\[
\frac{28x^7 + 40x^6 - 16x^4}{4x^3},
\]
we will divide each term in the numerator by \(4x^3\).

### Step 1: Divide each term

1. **First term:**
\[
\frac{28x^7}{4x^3} = \frac{28}{4} x^{7-3} = 7x^4
\]

2. **Second term:**
\[
\frac{40x^6}{4x^3} = \frac{40}{4} x^{6-3} = 10x^3
\]

3. **Third term:**
\[
\frac{-16x^4}{4x^3} = \frac{-16}{4} x^{4-3} = -4x
\]

### Step 2: Combine the results
Now, putting all of these together, we have:

\[
7x^4 + 10x^3 - 4x
\]

### Final Answer:
So the quotient in the form \( ?x^? + ?x^? - ?x \) is:
\[
7x^4 + 10x^3 - 4x
\]

Filling in the blanks:
- The first term is \(7x^4\), so \( ? = 7 \) and \( 4 \).
- The second term is \(10x^3\), so \( ? = 10 \) and \( 3 \).
- The last term is \(-4x\), so \( ? = 4 \).

Thus, the final representation is:

**7x^4 + 10x^3 - 4x.**