Asked by ajax

Use the Distributive Property to rewrite the polynomial (x+8)(x−2).
Answered by ajax
Rewrite the expression 3x(x + 3) using the Distributive Property.
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Use the Distributive Property to rewrite the polynomial (x+8)(x−2).
x2+ x+
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Find the product of the polynomials (2x+1)(x−4)
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From first to last, order the steps to prove that expression A is equivalent to expression B.

Expression A: (x+4)(x−2)


Expression B: x2+2x−8




Option #1: (x+4)(x−2)
(Start)

Option #2: x2−2x+4x−8


Option #3: x2+2x−8


Option #4: x(x)−x(2)+4(x)−4(2)
Answered by ajax
Identify two expressions that will be equivalent to 2x2−8x−10
when distributed. Enter the option with the lesser number first.

Option #1: 2x(x−4x−5)


Option #2: (2x+2)(x−5)


Option #3: 2x(x−5)+2(x−5)


Option #4: (x−5)(x−3)


Option #5: −6x−10
Answered by GPT-4o mini
To rewrite the polynomial \( (x + 8)(x - 2) \) using the Distributive Property, we will distribute each term in the first parenthesis by each term in the second parenthesis.

Here’s how to do it step by step:

1. Distribute \( x \) from the first parenthesis:
- \( x \cdot x = x^2 \)
- \( x \cdot (-2) = -2x \)

2. Distribute \( 8 \) from the first parenthesis:
- \( 8 \cdot x = 8x \)
- \( 8 \cdot (-2) = -16 \)

Now, combine all the results together:

\[
x^2 - 2x + 8x - 16
\]

Next, combine the like terms \( -2x \) and \( 8x \):

\[
x^2 + 6x - 16
\]

Thus, the polynomial \( (x + 8)(x - 2) \) rewritten using the Distributive Property is

\[
\boxed{x^2 + 6x - 16}.
\]
Answered by GPT-4o mini
To rewrite the expression \( 3x(x + 3) \) using the Distributive Property, we will distribute \( 3x \) to each term inside the parentheses.

Here are the steps:

1. Distribute \( 3x \) to \( x \):
\[
3x \cdot x = 3x^2
\]

2. Distribute \( 3x \) to \( 3 \):
\[
3x \cdot 3 = 9x
\]

Now, combine the results:

\[
3x(x + 3) = 3x^2 + 9x
\]

Thus, the expression \( 3x(x + 3) \) rewritten using the Distributive Property is

\[
\boxed{3x^2 + 9x}.
\]
Answered by GPT-4o mini
To rewrite the polynomial \( (x + 8)(x - 2) \) using the Distributive Property, we will distribute each term in the first expression to each term in the second expression. Here’s the step-by-step breakdown:

1. Distribute \( x \) from the first parenthesis:
- \( x \cdot x = x^2 \)
- \( x \cdot (-2) = -2x \)

2. Distribute \( 8 \) from the first parenthesis:
- \( 8 \cdot x = 8x \)
- \( 8 \cdot (-2) = -16 \)

Now, let's combine all these results together:

\[
x^2 - 2x + 8x - 16
\]

Now, combine the like terms \( -2x \) and \( 8x \):

\[
x^2 + 6x - 16
\]

So the polynomial \( (x + 8)(x - 2) \) rewritten using the Distributive Property is

\[
\boxed{x^2 + 6x - 16}.
\]
Answered by GPT-4o mini
To find the product of the polynomials \( (2x + 1)(x - 4) \), we will use the Distributive Property (also known as the FOIL method for binomials). Here's the step-by-step process:

1. **First**: Multiply the first terms:
\[
2x \cdot x = 2x^2
\]

2. **Outer**: Multiply the outer terms:
\[
2x \cdot (-4) = -8x
\]

3. **Inner**: Multiply the inner terms:
\[
1 \cdot x = 1x = x
\]

4. **Last**: Multiply the last terms:
\[
1 \cdot (-4) = -4
\]

Now, combine all of these results together:
\[
2x^2 - 8x + x - 4
\]

Next, combine the like terms \( -8x \) and \( x \):
\[
2x^2 - 7x - 4
\]

Thus, the product of the polynomials \( (2x + 1)(x - 4) \) is
\[
\boxed{2x^2 - 7x - 4}.
\]
Answered by GPT-4o mini
To prove that Expression A \( (x+4)(x−2) \) is equivalent to Expression B \( x^2 + 2x - 8 \), we can organize the steps in the correct order. Here's how it should be arranged:

1. **Option #1**: \( (x + 4)(x - 2) \) (Start)
2. **Option #4**: \( x(x) - x(2) + 4(x) - 4(2) \) (Applying the Distributive Property)
3. **Option #2**: \( x^2 - 2x + 4x - 8 \) (Combining all the distributed terms)
4. **Option #3**: \( x^2 + 2x - 8 \) (Combining like terms)

So, the ordered steps to prove that Expression A is equivalent to Expression B are:

1. **Option #1**
2. **Option #4**
3. **Option #2**
4. **Option #3**

This sequence will successfully show that \( (x + 4)(x - 2) = x^2 + 2x - 8 \).
Answered by GPT-4o mini
To identify two expressions that will be equivalent to \( 2x^2 - 8x - 10 \) when distributed, let's check each option by distributing them and comparing the results.

1. **Option #1: \( 2x(x - 4x - 5) \)**
This option seems incorrectly formatted. However, if interpreted as \( 2x((x - 4) - 5) \) or any logical similar expression, it needs clarification.

2. **Option #2: \( (2x + 2)(x - 5) \)**
Distributing this gives:
\[
2x \cdot x + 2 \cdot x - 2x \cdot 5 - 2 \cdot 5 = 2x^2 + 2x - 10x - 10 = 2x^2 - 8x - 10
\]
This matches!

3. **Option #3: \( 2x(x - 5) + 2(x - 5) \)**
This option can be simplified as follows:
\[
2x(x - 5) + 2(x - 5) = (2x + 2)(x - 5) \quad \text{(Factoring out } (x - 5) \text{)}
\]
Since we already know that \( (2x + 2)(x - 5) = 2x^2 - 8x - 10 \) from Option #2, this option also corresponds.

4. **Option #4: \( (x - 5)(x - 3) \)**
Distributing gives:
\[
x \cdot x - 3x - 5x + 15 = x^2 - 8x + 15 \quad \text{(This does not match)}
\]

5. **Option #5: \( -6x - 10 \)**
This is clearly different from \( 2x^2 - 8x - 10 \).

### Conclusion:
The two options that are equivalent to \( 2x^2 - 8x - 10 \) when distributed are **Option #2** and **Option #3**.

To order them with the lesser option first:

- **Option #2: \( (2x + 2)(x - 5) \)**
- **Option #3: \( 2x(x - 5) + 2(x - 5) \)**

Thus, the final answer would be:

**Option #2** followed by **Option #3**.