Question
Use the Distributive Property to rewrite the polynomial (x + 8)(x - 2)
__x^2+__x+__
__x^2+__x+__
Answers
Answer
Rewrite the expression 3x(x+3) using the Distributive Property
Answer
Find the product of the polynomials (2x + 1)(x - 4)
__x^2+__x+__
__x^2+__x+__
Answer
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: x^2+2x-8
Option #1: (x+4)(x-2)
Option #2: x^2-2x+4x-8
Option #3: x^2 + 2x-8
Option #4: x(x)-x(2)+4(x)-4(2)
First: Option #__
Second: Option # __
Third: Option #__
Fourth: Option # __
Expression A: (x+4)(x-2)
Expression B: x^2+2x-8
Option #1: (x+4)(x-2)
Option #2: x^2-2x+4x-8
Option #3: x^2 + 2x-8
Option #4: x(x)-x(2)+4(x)-4(2)
First: Option #__
Second: Option # __
Third: Option #__
Fourth: Option # __
Answer
Identify two expressions that will be equivalent to 2x^2 - 8x - 10 when distributed. Enter the option with 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
Option#__ and Option# __
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
Option#__ and Option# __
Answered by
GPT-4o mini
To rewrite the polynomial \((x + 8)(x - 2)\) using the Distributive Property, we need to distribute each term in the first polynomial across each term in the second polynomial.
Here's how it works step-by-step:
1. Distribute \(x\) in \((x + 8)\) to both \(x\) and \(-2\):
\[
x \cdot x + x \cdot (-2) = x^2 - 2x
\]
2. Distribute \(8\) in \((x + 8)\) to both \(x\) and \(-2\):
\[
8 \cdot x + 8 \cdot (-2) = 8x - 16
\]
3. Now, combine all these results together:
\[
x^2 - 2x + 8x - 16
\]
4. Combine the like terms:
\[
x^2 + 6x - 16
\]
So, the final polynomial is:
\[
x^2 + 6x - 16
\]
In the format you provided, the filled-in blanks would be:
**1, 6, -16**
Thus, \((x + 8)(x - 2) = x^2 + 6x - 16\).
Here's how it works step-by-step:
1. Distribute \(x\) in \((x + 8)\) to both \(x\) and \(-2\):
\[
x \cdot x + x \cdot (-2) = x^2 - 2x
\]
2. Distribute \(8\) in \((x + 8)\) to both \(x\) and \(-2\):
\[
8 \cdot x + 8 \cdot (-2) = 8x - 16
\]
3. Now, combine all these results together:
\[
x^2 - 2x + 8x - 16
\]
4. Combine the like terms:
\[
x^2 + 6x - 16
\]
So, the final polynomial is:
\[
x^2 + 6x - 16
\]
In the format you provided, the filled-in blanks would be:
**1, 6, -16**
Thus, \((x + 8)(x - 2) = x^2 + 6x - 16\).
Answered by
GPT-4o mini
To rewrite the expression \(3x(x + 3)\) using the Distributive Property, you need to distribute \(3x\) to both terms inside the parentheses.
Here's how to do it step by step:
1. Distribute \(3x\) to \(x\):
\[
3x \cdot x = 3x^2
\]
2. Distribute \(3x\) to \(3\):
\[
3x \cdot 3 = 9x
\]
3. Now, combine the results:
\[
3x(x + 3) = 3x^2 + 9x
\]
Thus, the expression \(3x(x + 3)\) can be rewritten as:
\[
3x^2 + 9x
\]
Here's how to do it step by step:
1. Distribute \(3x\) to \(x\):
\[
3x \cdot x = 3x^2
\]
2. Distribute \(3x\) to \(3\):
\[
3x \cdot 3 = 9x
\]
3. Now, combine the results:
\[
3x(x + 3) = 3x^2 + 9x
\]
Thus, the expression \(3x(x + 3)\) can be rewritten as:
\[
3x^2 + 9x
\]
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).
1. **Distribute \(2x\) to both terms in \((x - 4)\)**:
\[
2x \cdot x = 2x^2
\]
\[
2x \cdot (-4) = -8x
\]
2. **Distribute \(1\) to both terms in \((x - 4)\)**:
\[
1 \cdot x = x
\]
\[
1 \cdot (-4) = -4
\]
3. **Now, combine all these results together**:
\[
2x^2 - 8x + x - 4
\]
4. **Combine the like terms**:
\[
2x^2 - 7x - 4
\]
So, the product of the polynomials \((2x + 1)(x - 4)\) is:
\[
2x^2 - 7x - 4
\]
In the format you provided, the filled-in blanks would be:
**2, -7, -4**
Thus, \((2x + 1)(x - 4) = 2x^2 - 7x - 4\).
1. **Distribute \(2x\) to both terms in \((x - 4)\)**:
\[
2x \cdot x = 2x^2
\]
\[
2x \cdot (-4) = -8x
\]
2. **Distribute \(1\) to both terms in \((x - 4)\)**:
\[
1 \cdot x = x
\]
\[
1 \cdot (-4) = -4
\]
3. **Now, combine all these results together**:
\[
2x^2 - 8x + x - 4
\]
4. **Combine the like terms**:
\[
2x^2 - 7x - 4
\]
So, the product of the polynomials \((2x + 1)(x - 4)\) is:
\[
2x^2 - 7x - 4
\]
In the format you provided, the filled-in blanks would be:
**2, -7, -4**
Thus, \((2x + 1)(x - 4) = 2x^2 - 7x - 4\).
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 follow these ordered steps:
1. **Start with Expression A**: \((x + 4)(x - 2)\)
- This is Option #1.
2. **Apply the Distributive Property (or FOIL method)** to expand:
\[
(x + 4)(x - 2) \implies x(x) + x(-2) + 4(x) + 4(-2) \implies x^2 - 2x + 4x - 8
\]
- This corresponds to Option #4.
3. **Combine like terms**:
\[
x^2 - 2x + 4x - 8 \implies x^2 + 2x - 8
\]
- This is Option #2.
4. **Now you have Expression B**: \(x^2 + 2x - 8\)
- This corresponds to Option #3.
Putting it all together, the ordered steps are:
- First: Option #1
- Second: Option #4
- Third: Option #2
- Fourth: Option #3
So, the final answer is:
- First: Option #1
- Second: Option #4
- Third: Option #2
- Fourth: Option #3
1. **Start with Expression A**: \((x + 4)(x - 2)\)
- This is Option #1.
2. **Apply the Distributive Property (or FOIL method)** to expand:
\[
(x + 4)(x - 2) \implies x(x) + x(-2) + 4(x) + 4(-2) \implies x^2 - 2x + 4x - 8
\]
- This corresponds to Option #4.
3. **Combine like terms**:
\[
x^2 - 2x + 4x - 8 \implies x^2 + 2x - 8
\]
- This is Option #2.
4. **Now you have Expression B**: \(x^2 + 2x - 8\)
- This corresponds to Option #3.
Putting it all together, the ordered steps are:
- First: Option #1
- Second: Option #4
- Third: Option #2
- Fourth: Option #3
So, the final answer is:
- First: Option #1
- Second: Option #4
- Third: Option #2
- Fourth: Option #3
Answered by
GPT-4o mini
To identify the two expressions that will be equivalent to \(2x^2 - 8x - 10\) when distributed, let’s analyze each option:
### Check Each Option:
1. **Option #1: \(2x(x - 4x - 5)\)**
- Incorrect expression as it has an improperly formatted parentheses.
2. **Option #2: \((2x + 2)(x - 5)\)**
- Distributing gives:
\[
(2x + 2)(x - 5) = 2x(x) + 2(x) - 2x(5) - 2(5) = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10
\]
- This matches our target expression!
3. **Option #3: \(2x(x - 5) + 2(x - 5)\)**
- Distributing gives:
\[
2x(x - 5) + 2(x - 5) = (2x^2 - 10x) + (2x - 10) = 2x^2 - 8x - 10
\]
- This matches our target expression!
4. **Option #4: \((x - 5)(x - 3)\)**
- Distributing gives:
\[
(x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15
\]
- This does not match our target expression.
5. **Option #5: \(-6x - 10\)**
- This is a linear expression and does not contain the \(x^2\) term, so it does not match.
### Summary of Valid Options:
- Option #2: \((2x + 2)(x - 5)\)
- Option #3: \(2x(x - 5) + 2(x - 5)\)
### Final Answer:
**Option #2 and Option #3**
So the answer is:
- Option #2 and Option #3
### Check Each Option:
1. **Option #1: \(2x(x - 4x - 5)\)**
- Incorrect expression as it has an improperly formatted parentheses.
2. **Option #2: \((2x + 2)(x - 5)\)**
- Distributing gives:
\[
(2x + 2)(x - 5) = 2x(x) + 2(x) - 2x(5) - 2(5) = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10
\]
- This matches our target expression!
3. **Option #3: \(2x(x - 5) + 2(x - 5)\)**
- Distributing gives:
\[
2x(x - 5) + 2(x - 5) = (2x^2 - 10x) + (2x - 10) = 2x^2 - 8x - 10
\]
- This matches our target expression!
4. **Option #4: \((x - 5)(x - 3)\)**
- Distributing gives:
\[
(x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15
\]
- This does not match our target expression.
5. **Option #5: \(-6x - 10\)**
- This is a linear expression and does not contain the \(x^2\) term, so it does not match.
### Summary of Valid Options:
- Option #2: \((2x + 2)(x - 5)\)
- Option #3: \(2x(x - 5) + 2(x - 5)\)
### Final Answer:
**Option #2 and Option #3**
So the answer is:
- Option #2 and Option #3
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