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Fill in the blanks to complete the polynomial equation that shows the subtraction of the second expression from the first and t...Asked by x
Fill in the blanks to complete the polynomial equation that shows the subtraction of the second expression from the first and the resulting difference. (2 points)
9x^2 + ( ) + 13
- 2x^2 + 12x + ( )
( )x^2 + 5x + 2
9x^2 + ( ) + 13
- 2x^2 + 12x + ( )
( )x^2 + 5x + 2
Answers
Answered by
x
Simplify (1/4x^2−3x+2.25)+(0.75x^2+2x−1/4)
Answered by
x
Add the polynomial expressions 3−2p−5p^2 and p^4−3p+4.(1 point)
Responses
p^4−5p^2−5p+7
p superscript 4 baseline minus 5 p squared minus 5 p plus 7
−5p^2−4p+7
negative 5 p squared minus 4 p plus 7
p^4−5p^2−6p+7
p superscript 4 baseline minus 5 p squared minus 6 p plus 7
−4p^2−5p+7
Responses
p^4−5p^2−5p+7
p superscript 4 baseline minus 5 p squared minus 5 p plus 7
−5p^2−4p+7
negative 5 p squared minus 4 p plus 7
p^4−5p^2−6p+7
p superscript 4 baseline minus 5 p squared minus 6 p plus 7
−4p^2−5p+7
Answered by
x
What is (3x^2−2)+(11−5x) ?(1 point)
Responses
−2x^2+9
negative 2 x squared plus 9
3x^2−5x+9
3 x squared minus 5 x plus 9
3x^2+5x−9
3 x squared plus 5 x minus 9
2x^2−9
Responses
−2x^2+9
negative 2 x squared plus 9
3x^2−5x+9
3 x squared minus 5 x plus 9
3x^2+5x−9
3 x squared plus 5 x minus 9
2x^2−9
Answered by
x
Which example correctly demonstrates polynomial addition?(1 point)
Responses
(2x2+6x+1)+(3x2+3x+9)=13x2+9x+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals left parenthesis 2 x squared plus 3 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 6 x plus 9 right parenthesis
(2x2+6x+1)+(3x2+3x+9)= 5x4+9x2+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals left parenthesis 2 x squared plus 3 x squared right parenthesis plus left parenthesis 6 x plus 3 x right parenthesis plus left parenthesis 1 plus 9 right parenthesis
(2x2+6x+1)+(3x2+3x+9)=5x2+9x+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals 5 x squared plus 9 x plus 10
(2x2+6x+1)+(3x2+3x+9)=−x2+3x−8
Responses
(2x2+6x+1)+(3x2+3x+9)=13x2+9x+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals left parenthesis 2 x squared plus 3 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 6 x plus 9 right parenthesis
(2x2+6x+1)+(3x2+3x+9)= 5x4+9x2+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals left parenthesis 2 x squared plus 3 x squared right parenthesis plus left parenthesis 6 x plus 3 x right parenthesis plus left parenthesis 1 plus 9 right parenthesis
(2x2+6x+1)+(3x2+3x+9)=5x2+9x+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals 5 x squared plus 9 x plus 10
(2x2+6x+1)+(3x2+3x+9)=−x2+3x−8
Answered by
x
Which example correctly demonstrates polynomial subtraction?(1 point)
Responses
(2x2+6x+1)−(3x2+3x+9)=−x2+3x−8
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis minus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals negative x squared plus 3 x minus 8
(2x2+6x+1)−(3x2+3x+9)=5x2+9x+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis minus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals 5 x squared plus 9 x plus 10
(2x2+6x+1)−(3x2+3x+9)=x2+3x+8
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis minus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals left parenthesis 2 x squared plus 6 x minus 4 right parenthesis minus left parenthesis 3 x squared plus 3 x plus 4 right parenthesis
(2x2+6x+1)−(3x2+3x+9)=−5x2+3x−8
Responses
(2x2+6x+1)−(3x2+3x+9)=−x2+3x−8
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis minus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals negative x squared plus 3 x minus 8
(2x2+6x+1)−(3x2+3x+9)=5x2+9x+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis minus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals 5 x squared plus 9 x plus 10
(2x2+6x+1)−(3x2+3x+9)=x2+3x+8
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis minus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals left parenthesis 2 x squared plus 6 x minus 4 right parenthesis minus left parenthesis 3 x squared plus 3 x plus 4 right parenthesis
(2x2+6x+1)−(3x2+3x+9)=−5x2+3x−8
Answered by
GPT-4o mini
To fill in the blanks for the polynomial equation resulting from the subtraction, let's compare the expressions given:
1. The first expression: \( 9x^2 + ( \, ? \, ) + 13 \)
2. The second expression (to be subtracted): \( - (2x^2 + 12x + ( \, ? \, )) \)
To subtract the second expression from the first, we need to subtract each corresponding term:
1. For the \(x^2\) terms:
\( 9x^2 - 2x^2 = (9 - 2)x^2 = 7x^2\)
2. For the \(x\) terms:
We need to determine what the first expression's \(x\) coefficient should be to leave a difference of \(5x\). Since we have \( -12x \) when the second expression is subtracted, we can set up the equation:
\[ ( \, ? \, ) - 12 = 5 \]
Solving for \( ? \):
\[ ( \, ? \, ) = 5 + 12 = 17 \]
3. For the constant terms:
We need to determine what the first expression's constant term should be to leave a difference of \(2\):
\[ 13 - ( \, ? \, ) = 2 \]
Solving for \( ? \):
\[ ( \, ? \, ) = 13 - 2 = 11 \]
Now we can fill the blanks:
1. The first blank (x term) should be: **17**
2. The second blank (constant term) should be: **11**
Putting it all together, the completed polynomial equation shows:
\[
9x^2 + 17x + 13
- (2x^2 + 12x + 11)
= 7x^2 + 5x + 2
\]
So the final answers for the blanks are:
**First Blank:** 17
**Second Blank:** 11
**Third Blank:** 7
The output will look like:
9x² + (17) + 13
- 2x² + 12x + (11)
(7)x² + 5x + 2
1. The first expression: \( 9x^2 + ( \, ? \, ) + 13 \)
2. The second expression (to be subtracted): \( - (2x^2 + 12x + ( \, ? \, )) \)
To subtract the second expression from the first, we need to subtract each corresponding term:
1. For the \(x^2\) terms:
\( 9x^2 - 2x^2 = (9 - 2)x^2 = 7x^2\)
2. For the \(x\) terms:
We need to determine what the first expression's \(x\) coefficient should be to leave a difference of \(5x\). Since we have \( -12x \) when the second expression is subtracted, we can set up the equation:
\[ ( \, ? \, ) - 12 = 5 \]
Solving for \( ? \):
\[ ( \, ? \, ) = 5 + 12 = 17 \]
3. For the constant terms:
We need to determine what the first expression's constant term should be to leave a difference of \(2\):
\[ 13 - ( \, ? \, ) = 2 \]
Solving for \( ? \):
\[ ( \, ? \, ) = 13 - 2 = 11 \]
Now we can fill the blanks:
1. The first blank (x term) should be: **17**
2. The second blank (constant term) should be: **11**
Putting it all together, the completed polynomial equation shows:
\[
9x^2 + 17x + 13
- (2x^2 + 12x + 11)
= 7x^2 + 5x + 2
\]
So the final answers for the blanks are:
**First Blank:** 17
**Second Blank:** 11
**Third Blank:** 7
The output will look like:
9x² + (17) + 13
- 2x² + 12x + (11)
(7)x² + 5x + 2
Answered by
GPT-4o mini
To simplify the expression \((\frac{1}{4}x^2 - 3x + 2.25) + (0.75x^2 + 2x - \frac{1}{4})\), we will first combine like terms.
### Step 1: Rewrite the expression clearly
\[
\left(\frac{1}{4}x^2 - 3x + 2.25\right) + \left(0.75x^2 + 2x - \frac{1}{4}\right)
\]
### Step 2: Combine the terms
Let's combine the \(x^2\), \(x\), and constant terms.
#### Combine the \(x^2\) terms:
1. \(\frac{1}{4}x^2 + 0.75x^2\)
To combine these, convert \(0.75\) into a fraction:
\[
0.75 = \frac{3}{4}
\]
Now add:
\[
\frac{1}{4}x^2 + \frac{3}{4}x^2 = \left(\frac{1 + 3}{4}\right)x^2 = \frac{4}{4}x^2 = 1x^2 = x^2
\]
#### Combine the \(x\) terms:
2. \(-3x + 2x = (-3 + 2)x = -1x = -x\)
#### Combine the constant terms:
3. \(2.25 - \frac{1}{4}\)
Convert \(2.25\) into a fraction:
\[
2.25 = \frac{9}{4}
\]
Now, subtract:
\[
\frac{9}{4} - \frac{1}{4} = \frac{9 - 1}{4} = \frac{8}{4} = 2
\]
### Step 3: Combine all the simplified parts
Now we can put it all together:
\[
x^2 - x + 2
\]
### Final simplified expression:
Thus, the simplified expression is:
\[
\boxed{x^2 - x + 2}
\]
### Step 1: Rewrite the expression clearly
\[
\left(\frac{1}{4}x^2 - 3x + 2.25\right) + \left(0.75x^2 + 2x - \frac{1}{4}\right)
\]
### Step 2: Combine the terms
Let's combine the \(x^2\), \(x\), and constant terms.
#### Combine the \(x^2\) terms:
1. \(\frac{1}{4}x^2 + 0.75x^2\)
To combine these, convert \(0.75\) into a fraction:
\[
0.75 = \frac{3}{4}
\]
Now add:
\[
\frac{1}{4}x^2 + \frac{3}{4}x^2 = \left(\frac{1 + 3}{4}\right)x^2 = \frac{4}{4}x^2 = 1x^2 = x^2
\]
#### Combine the \(x\) terms:
2. \(-3x + 2x = (-3 + 2)x = -1x = -x\)
#### Combine the constant terms:
3. \(2.25 - \frac{1}{4}\)
Convert \(2.25\) into a fraction:
\[
2.25 = \frac{9}{4}
\]
Now, subtract:
\[
\frac{9}{4} - \frac{1}{4} = \frac{9 - 1}{4} = \frac{8}{4} = 2
\]
### Step 3: Combine all the simplified parts
Now we can put it all together:
\[
x^2 - x + 2
\]
### Final simplified expression:
Thus, the simplified expression is:
\[
\boxed{x^2 - x + 2}
\]
Answered by
GPT-4o mini
To add the polynomial expressions \(3 - 2p - 5p^2\) and \(p^4 - 3p + 4\), we proceed as follows:
### Step 1: Write the polynomials clearly:
1. First polynomial: \(3 - 2p - 5p^2\)
2. Second polynomial: \(p^4 - 3p + 4\)
### Step 2: Combine the two polynomials:
We can rewrite the addition:
\[
(3 - 2p - 5p^2) + (p^4 - 3p + 4)
\]
### Step 3: Group like terms:
Now group the terms with the same powers of \(p\):
- \(p^4\) terms: \(1p^4\)
- \(p^2\) terms: \(-5p^2\)
- \(p\) terms: \(-2p - 3p = -5p\)
- constant terms: \(3 + 4 = 7\)
### Step 4: Write the combined polynomial:
Now we can combine everything into a single polynomial:
\[
p^4 - 5p^2 - 5p + 7
\]
### Conclusion:
So the final result is:
\[
\boxed{p^4 - 5p^2 - 5p + 7}
\]
This matches with the response option:
\[
p^4 - 5p^2 - 5p + 7
\]
### Step 1: Write the polynomials clearly:
1. First polynomial: \(3 - 2p - 5p^2\)
2. Second polynomial: \(p^4 - 3p + 4\)
### Step 2: Combine the two polynomials:
We can rewrite the addition:
\[
(3 - 2p - 5p^2) + (p^4 - 3p + 4)
\]
### Step 3: Group like terms:
Now group the terms with the same powers of \(p\):
- \(p^4\) terms: \(1p^4\)
- \(p^2\) terms: \(-5p^2\)
- \(p\) terms: \(-2p - 3p = -5p\)
- constant terms: \(3 + 4 = 7\)
### Step 4: Write the combined polynomial:
Now we can combine everything into a single polynomial:
\[
p^4 - 5p^2 - 5p + 7
\]
### Conclusion:
So the final result is:
\[
\boxed{p^4 - 5p^2 - 5p + 7}
\]
This matches with the response option:
\[
p^4 - 5p^2 - 5p + 7
\]
Answered by
GPT-4o mini
To simplify the expression \((3x^2 - 2) + (11 - 5x)\), we will combine like terms.
### Step 1: Write the expression clearly
\[
(3x^2 - 2) + (11 - 5x)
\]
### Step 2: Remove parentheses
We can rewrite the expression by removing the parentheses:
\[
3x^2 - 2 + 11 - 5x
\]
### Step 3: Combine like terms
Now, combine the constant terms and the \(x\) terms:
- The \(x^2\) term: \(3x^2\)
- The \(x\) term: \(-5x\)
- The constant terms: \(-2 + 11 = 9\)
Putting it all together, we have:
\[
3x^2 - 5x + 9
\]
### Conclusion
The simplified expression is:
\[
\boxed{3x^2 - 5x + 9}
\]
This matches the response option:
**3x² - 5x + 9**.
### Step 1: Write the expression clearly
\[
(3x^2 - 2) + (11 - 5x)
\]
### Step 2: Remove parentheses
We can rewrite the expression by removing the parentheses:
\[
3x^2 - 2 + 11 - 5x
\]
### Step 3: Combine like terms
Now, combine the constant terms and the \(x\) terms:
- The \(x^2\) term: \(3x^2\)
- The \(x\) term: \(-5x\)
- The constant terms: \(-2 + 11 = 9\)
Putting it all together, we have:
\[
3x^2 - 5x + 9
\]
### Conclusion
The simplified expression is:
\[
\boxed{3x^2 - 5x + 9}
\]
This matches the response option:
**3x² - 5x + 9**.
Answered by
GPT-4o mini
To determine which example correctly demonstrates polynomial addition, we need to properly add the two given polynomials:
1. **First Polynomial:** \(2x^2 + 6x + 1\)
2. **Second Polynomial:** \(3x^2 + 3x + 9\)
### Step 1: Combine Like Terms
Adding the polynomials involves adding the coefficients of the like terms together:
- **For \(x^2\) terms:**
\[
2x^2 + 3x^2 = (2 + 3)x^2 = 5x^2
\]
- **For \(x\) terms:**
\[
6x + 3x = (6 + 3)x = 9x
\]
- **For constant terms:**
\[
1 + 9 = 10
\]
### Step 2: Write the Final Result
Combining all these results, we get:
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10
\]
### Step 3: Check the Given Options
Now we can compare the calculations we've made with the provided options:
1. **Option 1:**
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 13x^2 + 9x + 10 \quad \text{(Incorrect)}
\]
2. **Option 2:**
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^4 + 9x^2 + 10 \quad \text{(Incorrect)}
\]
3. **Option 3:**
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \quad \text{(Correct)}
\]
4. **Option 4:**
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8 \quad \text{(Incorrect)}
\]
### Conclusion
The option that correctly demonstrates polynomial addition is:
\[
\boxed{(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10}
\]
1. **First Polynomial:** \(2x^2 + 6x + 1\)
2. **Second Polynomial:** \(3x^2 + 3x + 9\)
### Step 1: Combine Like Terms
Adding the polynomials involves adding the coefficients of the like terms together:
- **For \(x^2\) terms:**
\[
2x^2 + 3x^2 = (2 + 3)x^2 = 5x^2
\]
- **For \(x\) terms:**
\[
6x + 3x = (6 + 3)x = 9x
\]
- **For constant terms:**
\[
1 + 9 = 10
\]
### Step 2: Write the Final Result
Combining all these results, we get:
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10
\]
### Step 3: Check the Given Options
Now we can compare the calculations we've made with the provided options:
1. **Option 1:**
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 13x^2 + 9x + 10 \quad \text{(Incorrect)}
\]
2. **Option 2:**
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^4 + 9x^2 + 10 \quad \text{(Incorrect)}
\]
3. **Option 3:**
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \quad \text{(Correct)}
\]
4. **Option 4:**
\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8 \quad \text{(Incorrect)}
\]
### Conclusion
The option that correctly demonstrates polynomial addition is:
\[
\boxed{(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10}
\]
Answered by
GPT-4o mini
To determine which example correctly demonstrates polynomial subtraction, we need to perform the subtraction of the two provided polynomials:
1. **First Polynomial:** \(2x^2 + 6x + 1\)
2. **Second Polynomial:** \(3x^2 + 3x + 9\)
### Step 1: Subtract Polynomials
Subtraction can be interpreted as distributing a negative sign through the second polynomial. Thus, we rewrite the operation as:
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = (2x^2 + 6x + 1) + (-3x^2 - 3x - 9).
\]
### Step 2: Combine Like Terms
Now we add the new polynomial:
\[
2x^2 + 6x + 1 - 3x^2 - 3x - 9
\]
Now, let's combine the like terms:
- **For \(x^2\) terms:**
\[
2x^2 - 3x^2 = -1x^2 = -x^2
\]
- **For \(x\) terms:**
\[
6x - 3x = 3x
\]
- **For constant terms:**
\[
1 - 9 = -8
\]
### Step 3: Write the Final Result
Putting it all together:
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8
\]
### Step 4: Check Against the Given Options
Now we will verify our result against the provided options:
1. **Option 1:**
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \quad \text{(Correct)}
\]
2. **Option 2:**
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \quad \text{(Incorrect)}
\]
3. **Option 3:**
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8 \quad \text{(Incorrect)}
\]
4. **Option 4:**
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8 \quad \text{(Incorrect)}
\]
### Conclusion
The option that correctly demonstrates polynomial subtraction is:
\[
\boxed{(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8}
\]
1. **First Polynomial:** \(2x^2 + 6x + 1\)
2. **Second Polynomial:** \(3x^2 + 3x + 9\)
### Step 1: Subtract Polynomials
Subtraction can be interpreted as distributing a negative sign through the second polynomial. Thus, we rewrite the operation as:
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = (2x^2 + 6x + 1) + (-3x^2 - 3x - 9).
\]
### Step 2: Combine Like Terms
Now we add the new polynomial:
\[
2x^2 + 6x + 1 - 3x^2 - 3x - 9
\]
Now, let's combine the like terms:
- **For \(x^2\) terms:**
\[
2x^2 - 3x^2 = -1x^2 = -x^2
\]
- **For \(x\) terms:**
\[
6x - 3x = 3x
\]
- **For constant terms:**
\[
1 - 9 = -8
\]
### Step 3: Write the Final Result
Putting it all together:
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8
\]
### Step 4: Check Against the Given Options
Now we will verify our result against the provided options:
1. **Option 1:**
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \quad \text{(Correct)}
\]
2. **Option 2:**
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \quad \text{(Incorrect)}
\]
3. **Option 3:**
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8 \quad \text{(Incorrect)}
\]
4. **Option 4:**
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8 \quad \text{(Incorrect)}
\]
### Conclusion
The option that correctly demonstrates polynomial subtraction is:
\[
\boxed{(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8}
\]
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