Question
Add the polynomial expression 3-2p-5p^2 and p^4-3p+4
Answers
Answered by
GPT-4o mini
To add the polynomial expressions \(3 - 2p - 5p^2\) and \(p^4 - 3p + 4\), we combine like terms.
1. The first polynomial is:
\[
3 - 2p - 5p^2
\]
2. The second polynomial is:
\[
p^4 - 3p + 4
\]
Now, let's add them together:
\[
(3 - 2p - 5p^2) + (p^4 - 3p + 4)
\]
We will now group the like terms:
- \(p^4\) term: \(p^4\)
- \(p^2\) term: \(-5p^2\)
- \(p\) terms: \(-2p - 3p = -5p\)
- Constant terms: \(3 + 4 = 7\)
Now, we can combine all of these:
\[
p^4 - 5p^2 - 5p + 7
\]
So, the result of adding the two polynomials is:
\[
\boxed{p^4 - 5p^2 - 5p + 7}
\]
1. The first polynomial is:
\[
3 - 2p - 5p^2
\]
2. The second polynomial is:
\[
p^4 - 3p + 4
\]
Now, let's add them together:
\[
(3 - 2p - 5p^2) + (p^4 - 3p + 4)
\]
We will now group the like terms:
- \(p^4\) term: \(p^4\)
- \(p^2\) term: \(-5p^2\)
- \(p\) terms: \(-2p - 3p = -5p\)
- Constant terms: \(3 + 4 = 7\)
Now, we can combine all of these:
\[
p^4 - 5p^2 - 5p + 7
\]
So, the result of adding the two polynomials is:
\[
\boxed{p^4 - 5p^2 - 5p + 7}
\]
Answer
What is (3x^2-2)+(11-5x)?
Answered by
GPT-4o mini
To add the expressions \( (3x^2 - 2) + (11 - 5x) \), we will combine like terms.
1. The first expression is:
\[
3x^2 - 2
\]
2. The second expression is:
\[
11 - 5x
\]
Now, we add the two expressions together:
\[
(3x^2 - 2) + (11 - 5x)
\]
We will group the like terms:
- The \(x^2\) term: \(3x^2\)
- The \(x\) term: \(-5x\)
- The constant terms: \(-2 + 11 = 9\)
Putting it all together:
\[
3x^2 - 5x + 9
\]
So, the result of adding the two expressions is:
\[
\boxed{3x^2 - 5x + 9}
\]
1. The first expression is:
\[
3x^2 - 2
\]
2. The second expression is:
\[
11 - 5x
\]
Now, we add the two expressions together:
\[
(3x^2 - 2) + (11 - 5x)
\]
We will group the like terms:
- The \(x^2\) term: \(3x^2\)
- The \(x\) term: \(-5x\)
- The constant terms: \(-2 + 11 = 9\)
Putting it all together:
\[
3x^2 - 5x + 9
\]
So, the result of adding the two expressions is:
\[
\boxed{3x^2 - 5x + 9}
\]
Answer
Which example correctly demonstrates polynomial addition?
A. (2x^2+6x+1)+(3x^2+3x+9)=5x^2+9x+10
B. (2x^2+6x+1)+(3x^2+3x+9)=5x^4+9x^2+10
C. (2x^2+6x+1)+(3x^2+3x+9)=13^2+9x+10
D. (2x^2+6x+1)+(3x^2+3x+9)=x^2+3x-8
A. (2x^2+6x+1)+(3x^2+3x+9)=5x^2+9x+10
B. (2x^2+6x+1)+(3x^2+3x+9)=5x^4+9x^2+10
C. (2x^2+6x+1)+(3x^2+3x+9)=13^2+9x+10
D. (2x^2+6x+1)+(3x^2+3x+9)=x^2+3x-8
Answered by
GPT-4o mini
To determine which example correctly demonstrates polynomial addition, we will perform the addition for the polynomial expressions given in each case.
Given polynomials:
1. \(2x^2 + 6x + 1\)
2. \(3x^2 + 3x + 9\)
Now, let's perform the addition:
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9)
\]
Combining like terms:
- \(x^2\) terms: \(2x^2 + 3x^2 = 5x^2\)
- \(x\) terms: \(6x + 3x = 9x\)
- Constant terms: \(1 + 9 = 10\)
So, the result of the addition is:
\[
5x^2 + 9x + 10
\]
Now we can evaluate each option:
**A**: \((2x^2+6x+1)+(3x^2+3x+9)=5x^2+9x+10\)
This is correct.
**B**: \((2x^2+6x+1)+(3x^2+3x+9)=5x^4+9x^2+10\)
This is incorrect (should be \(5x^2 + 9x + 10\), not \(5x^4\)).
**C**: \((2x^2+6x+1)+(3x^2+3x+9)=13^2+9x+10\)
This is incorrect (the expression \(13^2\) does not relate to the polynomial addition).
**D**: \((2x^2+6x+1)+(3x^2+3x+9)=x^2+3x-8\)
This is incorrect (the correct result is \(5x^2 + 9x + 10\)).
Thus, the only correct example is:
\[
\text{A: } (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10
\]
So the correct answer is **A**.
Given polynomials:
1. \(2x^2 + 6x + 1\)
2. \(3x^2 + 3x + 9\)
Now, let's perform the addition:
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9)
\]
Combining like terms:
- \(x^2\) terms: \(2x^2 + 3x^2 = 5x^2\)
- \(x\) terms: \(6x + 3x = 9x\)
- Constant terms: \(1 + 9 = 10\)
So, the result of the addition is:
\[
5x^2 + 9x + 10
\]
Now we can evaluate each option:
**A**: \((2x^2+6x+1)+(3x^2+3x+9)=5x^2+9x+10\)
This is correct.
**B**: \((2x^2+6x+1)+(3x^2+3x+9)=5x^4+9x^2+10\)
This is incorrect (should be \(5x^2 + 9x + 10\), not \(5x^4\)).
**C**: \((2x^2+6x+1)+(3x^2+3x+9)=13^2+9x+10\)
This is incorrect (the expression \(13^2\) does not relate to the polynomial addition).
**D**: \((2x^2+6x+1)+(3x^2+3x+9)=x^2+3x-8\)
This is incorrect (the correct result is \(5x^2 + 9x + 10\)).
Thus, the only correct example is:
\[
\text{A: } (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10
\]
So the correct answer is **A**.
Answer
Which example correctly demonstrates polynomial subtraction?
A.(2x^2+6x+1)-(3x^3+3x+9)=-x^2+3x-8
B.(2x^2+6x+1)-(3x^3+3x+9)=x^2+3x+8
C.(2x^2+6x+1)-(3x^3+3x+9)=-5x^2+3x-8
D.(2x^2+6x+1)-(3x^3+3x+9)=5x^2+9x+10
A.(2x^2+6x+1)-(3x^3+3x+9)=-x^2+3x-8
B.(2x^2+6x+1)-(3x^3+3x+9)=x^2+3x+8
C.(2x^2+6x+1)-(3x^3+3x+9)=-5x^2+3x-8
D.(2x^2+6x+1)-(3x^3+3x+9)=5x^2+9x+10
Answered by
GPT-4o mini
To determine which example correctly demonstrates polynomial subtraction, we need to perform the subtraction of the given polynomial expressions:
**Polynomials to subtract:**
1. \(2x^2 + 6x + 1\)
2. \(3x^3 + 3x + 9\)
The subtraction operation is:
\[
(2x^2 + 6x + 1) - (3x^3 + 3x + 9)
\]
Distributing the negative sign:
\[
= 2x^2 + 6x + 1 - 3x^3 - 3x - 9
\]
Now, let's combine like terms, starting from the highest degree:
- **Cubic term:** \(-3x^3\)
- **Quadratic term:** \(2x^2\)
- **Linear term:** \(6x - 3x = 3x\)
- **Constant term:** \(1 - 9 = -8\)
Putting this together, we have:
\[
-3x^3 + 2x^2 + 3x - 8
\]
Now we can evaluate each option:
**A**: \((2x^2 + 6x + 1) - (3x^3 + 3x + 9) = -x^2 + 3x - 8\)
This is incorrect (the cubic term should be \(-3x^3\)).
**B**: \((2x^2 + 6x + 1) - (3x^3 + 3x + 9) = x^2 + 3x + 8\)
This is incorrect (the cubic term should be \(-3x^3\)).
**C**: \((2x^2 + 6x + 1) - (3x^3 + 3x + 9) = -5x^2 + 3x - 8\)
This is incorrect (the cubic term should be \(-3x^3\) and the quadratic term is \(2x^2\), not \(-5x^2\)).
**D**: \((2x^2 + 6x + 1) - (3x^3 + 3x + 9) = 5x^2 + 9x + 10\)
This is incorrect (it misrepresents the result, which includes a cubic term).
None of the options correctly demonstrate the subtraction of the two polynomials. The correct result is:
\[
-3x^3 + 2x^2 + 3x - 8
\]
Thus, none of the options A, B, C, or D are correct for this polynomial subtraction.
**Polynomials to subtract:**
1. \(2x^2 + 6x + 1\)
2. \(3x^3 + 3x + 9\)
The subtraction operation is:
\[
(2x^2 + 6x + 1) - (3x^3 + 3x + 9)
\]
Distributing the negative sign:
\[
= 2x^2 + 6x + 1 - 3x^3 - 3x - 9
\]
Now, let's combine like terms, starting from the highest degree:
- **Cubic term:** \(-3x^3\)
- **Quadratic term:** \(2x^2\)
- **Linear term:** \(6x - 3x = 3x\)
- **Constant term:** \(1 - 9 = -8\)
Putting this together, we have:
\[
-3x^3 + 2x^2 + 3x - 8
\]
Now we can evaluate each option:
**A**: \((2x^2 + 6x + 1) - (3x^3 + 3x + 9) = -x^2 + 3x - 8\)
This is incorrect (the cubic term should be \(-3x^3\)).
**B**: \((2x^2 + 6x + 1) - (3x^3 + 3x + 9) = x^2 + 3x + 8\)
This is incorrect (the cubic term should be \(-3x^3\)).
**C**: \((2x^2 + 6x + 1) - (3x^3 + 3x + 9) = -5x^2 + 3x - 8\)
This is incorrect (the cubic term should be \(-3x^3\) and the quadratic term is \(2x^2\), not \(-5x^2\)).
**D**: \((2x^2 + 6x + 1) - (3x^3 + 3x + 9) = 5x^2 + 9x + 10\)
This is incorrect (it misrepresents the result, which includes a cubic term).
None of the options correctly demonstrate the subtraction of the two polynomials. The correct result is:
\[
-3x^3 + 2x^2 + 3x - 8
\]
Thus, none of the options A, B, C, or D are correct for this polynomial subtraction.
Answer
What does it mean for a set to be "closed"?(1 point) Responses An operation performed on members of that set will result in a member of the same set. An operation performed on members of that set will result in a member of the same set. Members of a closed set cannot be added, subtracted, multiplied, or divided. Members of a closed set cannot be added, subtracted, multiplied, or divided. The set has a finite number of members, as opposed to being infinitely large. The set has a finite number of members, as opposed to being infinitely large. An expression belonging to a closed set cannot be used to create an expression that is not of that set.
Answered by
GPT-4o mini
The correct response to define what it means for a set to be "closed" is:
**An operation performed on members of that set will result in a member of the same set.**
This means that if you take any elements from the set and perform a certain operation (like addition, multiplication, etc.), the result of that operation will also be an element of the same set. For example, the set of integers is closed under addition because the sum of any two integers is also an integer.
**An operation performed on members of that set will result in a member of the same set.**
This means that if you take any elements from the set and perform a certain operation (like addition, multiplication, etc.), the result of that operation will also be an element of the same set. For example, the set of integers is closed under addition because the sum of any two integers is also an integer.