Adding & Subtracting Polynomials Quick Check

Question

1 of 51 of 5 Items
Question
Add the polynomial expressions 3−2p−5p2 and p4−3p+4.(1 point)
Responses

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

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

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

−4p2−5p+7

Answer 1

To add the polynomial expressions \(3 - 2p - 5p^2\) and \(p^4 - 3p + 4\), we first need to rearrange them by their degree:

1. \(p^4 - 3p + 4\)
2. \( - 5p^2 - 2p + 3\)

Now, we perform the addition term by term:

• For \(p^4\), we have: \[ p^4 \]

• For \(p^3\), there are no \(p^3\) terms in either polynomial, so it contributes: \[ 0 \]

• For \(p^2\), we have: \[ -5p^2 \]

• For \(p^1\), we combine: \[ -2p - 3p = -5p \]

• For the constant term, we combine: \[ 4 + 3 = 7 \]

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

The correct response is:

\(p^4 - 5p^2 - 5p + 7\).

Answer 2

Adding & Subtracting Polynomials Quick Check
2 of 52 of 5 Items
Question
What is (3x2−2)+(11−5x)?(1 point)
Responses

3x2−5x+9
3 x squared minus 5 x plus 9

2x2−9
2 x squared minus 9

3x2+5x−9
3 x squared plus 5 x minus 9

−2x2+9
negative 2 x squared plus 9
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Answer 3

To add the polynomial expressions \((3x^2 - 2)\) and \((11 - 5x)\), we can rearrange and combine like terms:

1. Start with: \[ 3x^2 - 2 + 11 - 5x \]

2. Combine the constant terms: \[ -2 + 11 = 9 \]

3. Now we rewrite the expression, arranging the terms by degree: \[ 3x^2 - 5x + 9 \]

Thus, the result of \((3x^2 - 2) + (11 - 5x)\) is:

\(3x^2 - 5x + 9\).

Answer 4

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

(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
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
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

Answer 5

To determine which example correctly demonstrates polynomial addition, let's perform the polynomial addition for \((2x^2 + 6x + 1) + (3x^2 + 3x + 9)\):

1. Combine the \(x^2\) terms: \[ 2x^2 + 3x^2 = 5x^2 \]

2. Combine the \(x\) terms: \[ 6x + 3x = 9x \]

3. Combine the constant terms: \[ 1 + 9 = 10 \]

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

Now, let's check each response:

1. \( (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^4 + 9x^2 + 10 \)
   Incorrect: The degree of the resulting polynomial cannot be \(x^4\).

2. \( (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \)
   Correct: This matches our calculation.

3. \( (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8 \)
   Incorrect: The signs and coefficients do not match.

4. \( (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 13x^2 + 9x + 10 \)
   Incorrect: The degree of the \(x^2\) terms is incorrect.

The correct response that demonstrates polynomial addition is:

\( (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \).

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 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)=−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

Answer 7

To determine which example correctly demonstrates polynomial subtraction, we will perform the subtraction of \((2x^2 + 6x + 1) - (3x^2 + 3x + 9)\):

1. Rewrite the expression explicitly: \[ (2x^2 + 6x + 1) - (3x^2 + 3x + 9) \]

2. Distribute the negative sign to each term in the second polynomial: \[ 2x^2 + 6x + 1 - 3x^2 - 3x - 9 \]

3. Combine like terms:

   • For \(x^2\) terms: \[ 2x^2 - 3x^2 = -1x^2 , \text{(or simply } -x^2\text{)} \]
   • For \(x\) terms: \[ 6x - 3x = 3x \]
   • For constant terms: \[ 1 - 9 = -8 \]

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

Let's now check each response:

1. \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8 \)
   Incorrect: The coefficient of \(x^2\) is incorrect.

2. \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8 \)
   Incorrect: The signs and coefficients do not match.

3. \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \)
   Correct: This matches our calculation exactly.

4. \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \)
   Incorrect: The signs and coefficients do not match.

The correct response that demonstrates polynomial subtraction is:

\( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \).

Answer 8

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

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.

Members of a closed set cannot be added, subtracted, multiplied, or divided.
Members of a closed set cannot be added, subtracted, multiplied, or divided.

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.

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.
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Answer 9

The correct response that describes 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 when you perform operations (such as addition, subtraction, multiplication, or division) on elements within the set, the result will also be an element of that set. For example, the set of integers is closed under addition because adding any two integers will yield another integer.