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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 the resulting difference. (2 points)

9x^2 + ( ) + 13
- 2x^2 + 12x + ( )
( )x^2 + 5x + 2

11 answers

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
Simplify (1/4x^2−3x+2.25)+(0.75x^2+2x−1/4)

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:

  1. \(-3x + 2x = (-3 + 2)x = -1x = -x\)

Combine the constant terms:

  1. \(2.25 - \frac{1}{4}\)

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: \[ \boxed{x^2 - x + 2} \]

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

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: \[ p^4 - 5p^2 - 5p + 7 \]

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

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.

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

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: \[ \boxed{(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10} \]

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

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: \[ \boxed{(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8} \]