Question

Ahmed found the product of (2/3g2−6n)2
using the area model. His partial work is shown. What is the partial product that goes in the orange box? What is the partial product that goes in the blue box? What is the product of (23g2−6n)2
written in standard form?(3 points)

Answer 1

Consider the polynomial −4(34p−2n)+6n−(8n+p)
.

Which expression is equivalent to the given polynomial?

(1 point)
Responses

6n−3p
6 n minus 3 p

6n−4p
6 n minus 4 p

−10n−2p
negative 10 n minus 2 p

−10−3p

Answer 2

Consider the polynomial −4(3/4p−2n)+6n−(8n+p)
.

Which expression is equivalent to the given polynomial?

(1 point)
Responses

6n−3p
6 n minus 3 p

6n−4p
6 n minus 4 p

−10n−2p
negative 10 n minus 2 p

−10−3p

Answer 3

To find the product of the expression \( ( \frac{2}{3}g^2 - 6n )^2 \) using the area model, we can start by expressing it in a more familiar form.

We know that squaring a binomial can be done using the formula:

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

In this case, let:
- \( a = \frac{2}{3}g^2 \)
- \( b = 6n \)

Now we compute \( a^2 \), \( 2ab \), and \( b^2 \):

1. **Calculate \( a^2 \)**:
\[
a^2 = \left(\frac{2}{3}g^2\right)^2 = \frac{4}{9}g^4
\]

2. **Calculate \( b^2 \)**:
\[
b^2 = (6n)^2 = 36n^2
\]

3. **Calculate \( 2ab \)**:
\[
2ab = 2 \cdot \frac{2}{3}g^2 \cdot 6n = 2 \cdot \frac{12}{3}g^2 \cdot n = 8g^2n
\]

Now, substitute these values into the squared binomial formula:

\[
( \frac{2}{3}g^2 - 6n )^2 = a^2 - 2ab + b^2 = \frac{4}{9}g^4 - 8g^2n + 36n^2
\]

Now, we can fill in the orange and blue boxes based on what they represent:

- The **orange box** will contain \( a^2 \), which is \( \frac{4}{9}g^4 \).
- The **blue box** will contain \( b^2 \), which is \( 36n^2 \).

Finally, we combine everything to write the product in standard form:

\[
( \frac{2}{3}g^2 - 6n )^2 = \frac{4}{9}g^4 - 8g^2n + 36n^2
\]

So, to summarize:
- **Partial product in orange box**: \( \frac{4}{9}g^4 \)
- **Partial product in blue box**: \( 36n^2 \)
- **Product in standard form**: \( \frac{4}{9}g^4 - 8g^2n + 36n^2 \)

Answer 4

To simplify the polynomial expression \(-4(34p - 2n) + 6n - (8n + p)\), we can distribute and combine like terms step by step.

1. **Distribute \(-4\)** in the first term:
\[
-4(34p - 2n) = -4 \cdot 34p + (-4) \cdot (-2n) = -136p + 8n
\]

2. **Rewrite the expression**:
\[
-136p + 8n + 6n - (8n + p)
\]

3. **Distribute the negative in the third term**:
\[
- (8n + p) = -8n - p
\]
So now our expression becomes:
\[
-136p + 8n + 6n - 8n - p
\]

4. **Combine like terms**:
- For the \(n\) terms:
\[
8n + 6n - 8n = 6n
\]
- For the \(p\) terms:
\[
-136p - p = -137p
\]

5. **Combine the results**:
\[
6n - 137p
\]

Now, we can check the options given:

- **6n - 3p** (not equivalent)
- **6n - 4p** (not equivalent)
- **-10n - 2p** (not equivalent)
- **-10 - 3p** (not equivalent)

None of the provided options are equivalent to the simplified expression \( 6n - 137p \). There may have been an error in the transcription of options or in understanding the initial expression.

If you'd like, please provide additional context or verify the options, and I can assist further.

Answer 5

Let's simplify the polynomial expression \(-4\left(\frac{3}{4}p - 2n\right) + 6n - (8n + p)\) step by step.

1. **Distribute \(-4\)** to the terms inside the parentheses:
\[
-4\left(\frac{3}{4}p - 2n\right) = -4 \cdot \frac{3}{4}p + -4 \cdot (-2n) = -3p + 8n
\]

2. **Rewrite the expression**:
\[
-3p + 8n + 6n - (8n + p)
\]

3. **Distribute the negative sign in the third term**:
\[
-(8n + p) = -8n - p
\]
So our expression now looks like:
\[
-3p + 8n + 6n - 8n - p
\]

4. **Combine like terms**:
- For the \(n\) terms:
\[
8n + 6n - 8n = 6n
\]
- For the \(p\) terms:
\[
-3p - p = -4p
\]

5. **Combine the results**:
\[
6n - 4p
\]

Now, we can check the options provided:

1. **6n - 3p** (not equivalent)
2. **6n - 4p** (this is equivalent)
3. **-10n - 2p** (not equivalent)
4. **-10 - 3p** (not equivalent)

The correct answer is:
**6n - 4p**.