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Original Question
Multiply the polynomial expression (x+2)(x−4) .(2 points) x2+x+Question
Multiply the polynomial expression (x+2)(x−4)
.(2 points)
x2+
x+
.(2 points)
x2+
x+
Answers
Answer
Multiply to simplify the polynomial expression (r−1)(r2−2r+3)
.(2 points)
r3+
r2+
r+
.(2 points)
r3+
r2+
r+
Answer
Simplify the polynomial expression (xy−2)(x2+1)
.(2 points)
x3y+
x2+
xy+
.(2 points)
x3y+
x2+
xy+
Answer
Which of the following shows that polynomials form a closed system under multiplication?
Option #1: 3(2x+5)=6x+15
Option #2: 8x+7=5x+3x+1+6
Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24
Option #4: 2x12⋅3=6x−−√
(1 point)
Option #
Option #1: 3(2x+5)=6x+15
Option #2: 8x+7=5x+3x+1+6
Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24
Option #4: 2x12⋅3=6x−−√
(1 point)
Option #
Answer
Which of the following options correctly describes if the following polynomial forms a closed system under multiplication?
−3(5x+4y)=−15x−12y
Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.
Option #2: No, because the exponent of x is not a positive integer.
Option #3: No, because multiplying the polynomials resulted in subtraction.
(1 point)
Option #
−3(5x+4y)=−15x−12y
Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.
Option #2: No, because the exponent of x is not a positive integer.
Option #3: No, because multiplying the polynomials resulted in subtraction.
(1 point)
Option #
Answer
it was option #2
Answered by
GPT-4o mini
To multiply the polynomial expression \((x + 2)(x - 4)\), we can use the distributive property (also known as the FOIL method for binomials):
\[
(x + 2)(x - 4) = x \cdot x + x \cdot (-4) + 2 \cdot x + 2 \cdot (-4)
\]
Now, let's break it down step by step:
1. \(x \cdot x = x^2\)
2. \(x \cdot (-4) = -4x\)
3. \(2 \cdot x = 2x\)
4. \(2 \cdot (-4) = -8\)
Now, combine all these results:
\[
x^2 - 4x + 2x - 8
\]
Combine like terms:
\[
x^2 + (-4x + 2x) - 8 = x^2 - 2x - 8
\]
Thus, the final result of multiplying \((x + 2)(x - 4)\) is:
\[
x^2 - 2x - 8
\]
So, filling in the blanks:
- \(x^2 +\)
- \(-2x +\)
- \(-8\)
The complete answer is:
\[
x^2 - 2x - 8
\]
\[
(x + 2)(x - 4) = x \cdot x + x \cdot (-4) + 2 \cdot x + 2 \cdot (-4)
\]
Now, let's break it down step by step:
1. \(x \cdot x = x^2\)
2. \(x \cdot (-4) = -4x\)
3. \(2 \cdot x = 2x\)
4. \(2 \cdot (-4) = -8\)
Now, combine all these results:
\[
x^2 - 4x + 2x - 8
\]
Combine like terms:
\[
x^2 + (-4x + 2x) - 8 = x^2 - 2x - 8
\]
Thus, the final result of multiplying \((x + 2)(x - 4)\) is:
\[
x^2 - 2x - 8
\]
So, filling in the blanks:
- \(x^2 +\)
- \(-2x +\)
- \(-8\)
The complete answer is:
\[
x^2 - 2x - 8
\]
Answered by
GPT-4o mini
To multiply and simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we will use the distributive property.
1. Distribute \(r\) across \((r^2 - 2r + 3)\):
- \(r \cdot r^2 = r^3\)
- \(r \cdot (-2r) = -2r^2\)
- \(r \cdot 3 = 3r\)
So, from this part, we have:
\[
r^3 - 2r^2 + 3r
\]
2. Now, distribute \(-1\) across \((r^2 - 2r + 3)\):
- \(-1 \cdot r^2 = -r^2\)
- \(-1 \cdot (-2r) = 2r\)
- \(-1 \cdot 3 = -3\)
So, from this part, we have:
\[
-r^2 + 2r - 3
\]
3. Now, combine all the terms:
\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]
4. Combine like terms:
- For \(r^2\): \(-2r^2 - r^2 = -3r^2\)
- For \(r\): \(3r + 2r = 5r\)
Putting it all together, we get:
\[
r^3 - 3r^2 + 5r - 3
\]
Thus, the simplified expression is:
\[
r^3 - 3r^2 + 5r - 3
\]
So, filling in the blanks:
- \(r^3 +\)
- \(-3r^2 +\)
- \(5r - 3\)
The complete answer is:
\[
r^3 - 3r^2 + 5r - 3
\]
1. Distribute \(r\) across \((r^2 - 2r + 3)\):
- \(r \cdot r^2 = r^3\)
- \(r \cdot (-2r) = -2r^2\)
- \(r \cdot 3 = 3r\)
So, from this part, we have:
\[
r^3 - 2r^2 + 3r
\]
2. Now, distribute \(-1\) across \((r^2 - 2r + 3)\):
- \(-1 \cdot r^2 = -r^2\)
- \(-1 \cdot (-2r) = 2r\)
- \(-1 \cdot 3 = -3\)
So, from this part, we have:
\[
-r^2 + 2r - 3
\]
3. Now, combine all the terms:
\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]
4. Combine like terms:
- For \(r^2\): \(-2r^2 - r^2 = -3r^2\)
- For \(r\): \(3r + 2r = 5r\)
Putting it all together, we get:
\[
r^3 - 3r^2 + 5r - 3
\]
Thus, the simplified expression is:
\[
r^3 - 3r^2 + 5r - 3
\]
So, filling in the blanks:
- \(r^3 +\)
- \(-3r^2 +\)
- \(5r - 3\)
The complete answer is:
\[
r^3 - 3r^2 + 5r - 3
\]
Answered by
GPT-4o mini
To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will use the distributive property.
1. Distribute \(xy\) across \((x^2 + 1)\):
- \(xy \cdot x^2 = x^3y\)
- \(xy \cdot 1 = xy\)
From this part, we have:
\[
x^3y + xy
\]
2. Now, distribute \(-2\) across \((x^2 + 1)\):
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
From this part, we have:
\[
-2x^2 - 2
\]
3. Now, combine all the terms:
\[
x^3y + xy - 2x^2 - 2
\]
As a result, the simplified expression is:
\[
x^3y - 2x^2 + xy - 2
\]
Filling in the blanks:
- \(x^3y +\)
- \(-2x^2 +\)
- \(xy - 2\)
The complete answer can be presented as:
\[
x^3y - 2x^2 + xy - 2
\]
1. Distribute \(xy\) across \((x^2 + 1)\):
- \(xy \cdot x^2 = x^3y\)
- \(xy \cdot 1 = xy\)
From this part, we have:
\[
x^3y + xy
\]
2. Now, distribute \(-2\) across \((x^2 + 1)\):
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
From this part, we have:
\[
-2x^2 - 2
\]
3. Now, combine all the terms:
\[
x^3y + xy - 2x^2 - 2
\]
As a result, the simplified expression is:
\[
x^3y - 2x^2 + xy - 2
\]
Filling in the blanks:
- \(x^3y +\)
- \(-2x^2 +\)
- \(xy - 2\)
The complete answer can be presented as:
\[
x^3y - 2x^2 + xy - 2
\]
Answered by
GPT-4o mini
To determine which option demonstrates that polynomials form a closed system under multiplication, we need to find the option that shows the product of two polynomials resulting in another polynomial.
Let's evaluate each option:
**Option #1: \(3(2x+5)=6x+15\)**
- This shows the multiplication of a polynomial \((2x + 5)\) by a constant (which is acceptable), resulting in a polynomial. Although it shows a valid operation, it is not explicitly demonstrating multiplication of two polynomials.
**Option #2: \(8x+7=5x+3x+1+6\)**
- This option shows an equation and a simplification of a polynomial expression. It doesn’t involve multiplication or demonstrate the closure property.
**Option #3: \((2x^2−4)(3y+6)=6x^2y+12x^2−12y−24\)**
- This option shows the multiplication of two polynomials \((2x^2 - 4)\) and \((3y + 6)\), resulting in another polynomial. This clearly demonstrates closure under multiplication.
**Option #4: \(2x^{12} \cdot 3=6x^{-−sqrt}\)**
- This option is incorrect as it improperly represents polynomial terms, especially with the radical in the result. It does not demonstrate closure since the result is not a polynomial.
The correct answer is:
**Option #3.**
Let's evaluate each option:
**Option #1: \(3(2x+5)=6x+15\)**
- This shows the multiplication of a polynomial \((2x + 5)\) by a constant (which is acceptable), resulting in a polynomial. Although it shows a valid operation, it is not explicitly demonstrating multiplication of two polynomials.
**Option #2: \(8x+7=5x+3x+1+6\)**
- This option shows an equation and a simplification of a polynomial expression. It doesn’t involve multiplication or demonstrate the closure property.
**Option #3: \((2x^2−4)(3y+6)=6x^2y+12x^2−12y−24\)**
- This option shows the multiplication of two polynomials \((2x^2 - 4)\) and \((3y + 6)\), resulting in another polynomial. This clearly demonstrates closure under multiplication.
**Option #4: \(2x^{12} \cdot 3=6x^{-−sqrt}\)**
- This option is incorrect as it improperly represents polynomial terms, especially with the radical in the result. It does not demonstrate closure since the result is not a polynomial.
The correct answer is:
**Option #3.**
Answered by
GPT-4o mini
To determine whether the given polynomial \(−3(5x+4y)=−15x−12y\) demonstrates closure under multiplication, we need to evaluate the result of multiplying polynomials and whether that result is indeed a polynomial.
### Evaluation:
1. The expression given is \(−3(5x + 4y)\), where \(−3\) is a constant and \(5x + 4y\) is a polynomial (a linear polynomial).
2. The result of the multiplication is \(−15x − 12y\).
3. Both the original polynomial and the resulting expression are polynomials because they are expressed as sums of terms where the variables have non-negative integer exponents.
### Analysis of the Options:
**Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.**
- **This is correct.** The multiplication of the constant \(-3\) and the polynomial \(5x + 4y\) produces another polynomial \(−15x − 12y\).
**Option #2: No, because the exponent of x is not a positive integer.**
- This is **incorrect**. In the resulting polynomial \(−15x − 12y\), the exponent of \(x\) is \(1\), which is a positive integer.
**Option #3: No, because multiplying the polynomials resulted in subtraction.**
- This is **incorrect**. The result being a sum or a difference does not affect whether the polynomials form a closed system under multiplication. Polynomials can have addition and subtraction as operations and still be considered polynomials.
### Conclusion:
The correct choice is:
**Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.**
### Evaluation:
1. The expression given is \(−3(5x + 4y)\), where \(−3\) is a constant and \(5x + 4y\) is a polynomial (a linear polynomial).
2. The result of the multiplication is \(−15x − 12y\).
3. Both the original polynomial and the resulting expression are polynomials because they are expressed as sums of terms where the variables have non-negative integer exponents.
### Analysis of the Options:
**Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.**
- **This is correct.** The multiplication of the constant \(-3\) and the polynomial \(5x + 4y\) produces another polynomial \(−15x − 12y\).
**Option #2: No, because the exponent of x is not a positive integer.**
- This is **incorrect**. In the resulting polynomial \(−15x − 12y\), the exponent of \(x\) is \(1\), which is a positive integer.
**Option #3: No, because multiplying the polynomials resulted in subtraction.**
- This is **incorrect**. The result being a sum or a difference does not affect whether the polynomials form a closed system under multiplication. Polynomials can have addition and subtraction as operations and still be considered polynomials.
### Conclusion:
The correct choice is:
**Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.**
Answered by
GPT-4o mini
I apologize for the confusion. Let's carefully examine the reasoning behind the statement that the polynomial forms a closed system under multiplication and why Option #2 might be suggested.
The definition of a polynomial requires that its exponents be non-negative integers. The expression in question was:
\[
-3(5x + 4y) = -15x - 12y
\]
In this case, both \(x\) and \(y\) have non-negative integer exponents in the final result (the exponent of \(x\) is 1, which is indeed a positive integer), and therefore, the result is still a polynomial.
**Option #2 states:** "No, because the exponent of x is not a positive integer."
This reasoning seems to be incorrect because \(x\) has an exponent of 1 in the resulting polynomial, which is a positive integer. Therefore, the conclusion that was drawn for Option #2 is not accurate according to the definition of polynomials.
The correct conclusion should indeed align with **Option #1**, which states that multiplying the original polynomial indeed produces a polynomial.
If there is any additional context or reasoning you would like to provide for why **Option #2** was considered the right answer, please let me know, and I’d be happy to review it further. Thank you for your understanding!
The definition of a polynomial requires that its exponents be non-negative integers. The expression in question was:
\[
-3(5x + 4y) = -15x - 12y
\]
In this case, both \(x\) and \(y\) have non-negative integer exponents in the final result (the exponent of \(x\) is 1, which is indeed a positive integer), and therefore, the result is still a polynomial.
**Option #2 states:** "No, because the exponent of x is not a positive integer."
This reasoning seems to be incorrect because \(x\) has an exponent of 1 in the resulting polynomial, which is a positive integer. Therefore, the conclusion that was drawn for Option #2 is not accurate according to the definition of polynomials.
The correct conclusion should indeed align with **Option #1**, which states that multiplying the original polynomial indeed produces a polynomial.
If there is any additional context or reasoning you would like to provide for why **Option #2** was considered the right answer, please let me know, and I’d be happy to review it further. Thank you for your understanding!