Add the polynomial expressions 3−2p−5p2 and p4−3p+4.(1 point)

Question

Responses

−5p2−4p+7
negative 5 p squared minus 4 p plus 7

−4p2−5p+7
negative 4 p squared minus 5 p plus 7

p4−5p2−6p+7
p superscript 4 baseline minus 5 p squared minus 6 p plus 7

p4−5p2−5p+7
p superscript 4 baseline minus 5 p squared minus 5 p plus 7
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Answer 1

To add the polynomial expressions \(3 - 2p - 5p^2\) and \(p^4 - 3p + 4\), we combine like terms.:

1. Identify the terms from both polynomials:

   • First polynomial: \(3\), \(-2p\), \(-5p^2\)
   • Second polynomial: \(p^4\), \(-3p\), \(4\)

2. Rewrite the polynomials together: \[ (3 + 4) + (-2p - 3p) + (-5p^2) + p^4 \]

3. Combine like terms:

   • Constant terms: \(3 + 4 = 7\)
   • \(p\) terms: \(-2p - 3p = -5p\)
   • \(p^2\) terms: \(-5p^2\)
   • \(p^4\) term: \(p^4\)

Putting these together, we get: \[ p^4 - 5p^2 - 5p + 7 \]

Thus, the resulting polynomial expression is: \[ \boxed{p^4 - 5p^2 - 5p + 7} \]

This corresponds to the response:

• \(p^4 - 5p^2 - 5p + 7\)

Answer 2

What is (3x2−2)+(11−5x)?(1 point)
Responses

3x2−5x+9
3 x squared minus 5 x plus 9

−2x2+9
negative 2 x squared plus 9

2x2−9
2 x squared minus 9

3x2+5x−9

Answer 3

To simplify the expression \((3x^2 - 2) + (11 - 5x)\), we will combine like terms.

1. Identify the terms:

   • From the first polynomial: \(3x^2\) and \(-2\)
   • From the second polynomial: \(11\) and \(-5x\)

2. Rewrite the expression grouping the like terms: \[ 3x^2 + (-5x) + (-2 + 11) \]

3. Combine the like terms:

   • The \(x^2\) term: \(3x^2\) (no other \(x^2\) terms to combine with)
   • The \(x\) term: \(-5x\)
   • The constant terms: \(-2 + 11 = 9\)

Putting it all together, we get: \[ 3x^2 - 5x + 9 \]

Therefore, the final result is: \[ \boxed{3x^2 - 5x + 9} \]

This corresponds to the response:

• \(3x^2 - 5x + 9\)

Answer 4

Which example correctly demonstrates polynomial addition?(1 point)
Responses

(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)= 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)=−x2+3x−8
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 negative x squared plus 3 x minus 8

(2x2+6x+1)+(3x2+3x+9)=13x2+9x+10

Answer 5

To determine which example correctly demonstrates polynomial addition, we need to add the polynomial expressions \((2x^2 + 6x + 1)\) and \((3x^2 + 3x + 9)\).

Step 1: Add the polynomials

1. Identify the terms:

   • From the first polynomial: \(2x^2\), \(6x\), and \(1\)
   • From the second polynomial: \(3x^2\), \(3x\), and \(9\)

2. Combine like terms:

   • \(x^2\) terms: \(2x^2 + 3x^2 = 5x^2\)
   • \(x\) terms: \(6x + 3x = 9x\)
   • Constant terms: \(1 + 9 = 10\)

Step 2: Write the result

Putting it all together, we have: \[ (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \]

Conclusion

Now we can evaluate the provided options:

1. (2x² + 6x + 1) + (3x² + 3x + 9) = 5x² + 9x + 10 (CORRECT)

2. (2x² + 6x + 1) + (3x² + 3x + 9) = 5x⁴ + 9x² + 10 (INCORRECT)

   • The degree of \(x^4\) is incorrect, sum should have \(5x²\).

3. (2x² + 6x + 1) + (3x² + 3x + 9) = −x² + 3x − 8 (INCORRECT)

   • The expression does not match the addition.

4. (2x² + 6x + 1) + (3x² + 3x + 9) = 13x² + 9x + 10 (INCORRECT)

   • The \(x²\) term is incorrect as it should be \(5x²\).

Thus, the correct response demonstrating polynomial addition is: \[ \boxed{(2x² + 6x + 1) + (3x² + 3x + 9) = 5x² + 9x + 10} \]

Answer 6

Which example correctly demonstrates polynomial subtraction?(1 point)
Responses

(2x2+6x+1)−(3x2+3x+9)=−5x2+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 2 x squared plus 6 x plus 1 minus 3 x squared minus 3 x minus 9

(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)=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+9x+10

Answer 7

To determine which example correctly demonstrates polynomial subtraction for the expressions \((2x^2 + 6x + 1) - (3x^2 + 3x + 9)\), let's perform the polynomial subtraction step-by-step.

Step 1: Rewrite the Expression

1. The expression is:
   \[ (2x^2 + 6x + 1) - (3x^2 + 3x + 9) \]

2. Distribute the negative sign through the second polynomial: \[ = 2x^2 + 6x + 1 - 3x^2 - 3x - 9 \]

Step 2: Combine Like Terms

Now we will combine like terms:

1. \(x^2\) terms:
   \(2x^2 - 3x^2 = -1x^2\) or \(-x^2\)

2. \(x\) terms:
   \(6x - 3x = 3x\)

3. Constant terms:
   \(1 - 9 = -8\)

Step 3: Write the Result

Putting it all together, we have: \[ (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \]

Conclusion

Now we can evaluate the provided options:

1. (2x² + 6x + 1) − (3x² + 3x + 9) = −5x² + 3x − 8 (INCORRECT)

   • The \(x^2\) term is incorrect; it should be \(-x^2\).

2. (2x² + 6x + 1) − (3x² + 3x + 9) = −x² + 3x − 8 (CORRECT)

3. (2x² + 6x + 1) − (3x² + 3x + 9) = x² + 3x + 8 (INCORRECT)

   • The \(x^2\) term and constant are incorrect.

4. (2x² + 6x + 1) − (3x² + 3x + 9) = 5x² + 9x + 10 (INCORRECT)

   • This expression represents addition, not subtraction.

Thus, the correct response demonstrating polynomial subtraction is: \[ \boxed{(2x² + 6x + 1) - (3x² + 3x + 9) = -x² + 3x - 8} \]

Answer 8

What does it mean for a set to be "closed"?(1 point)
Responses

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.
An expression belonging to a closed set cannot be used to create an expression that is not of that set.

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.

Answer 9

In mathematical terms, a set is said to be "closed" under a particular operation if, when that operation is performed on members of the set, the result is also a member of the same set.

Thus, the correct response 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 two (or more) elements in the set and perform the operation (like addition, multiplication, etc.), the result will still be an element of the set.

Therefore, the answer is: \[ \boxed{\text{An operation performed on members of that set will result in a member of the same set.}} \]