Question

Find the sum. (4p−2p2+8)+(5p2+2p)+(7−3p)
(
4
𝑝
−
2
𝑝
2
+
8
)
+
(
5
𝑝
2
+
2
𝑝
)
+
(
7
−
3
𝑝
)
(1 point)
Responses

3p2+6p+15
3
𝑝
2
+
6
𝑝
+
15
3 p squared plus 6 p plus 15

−2p2+8p+15
−
2
𝑝
2
+
8
𝑝
+
15
negative 2 p squared plus 8 p plus 15

3p2+3p+15
3
𝑝
2
+
3
𝑝
+
15
3 p squared plus 3 p plus 15

5p2+p+15

Answer 1

Which of the following shows that polynomials are a closed system under addition?(1 point)
Responses

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=3.903x2−12.84x+3.903
(
0.923
𝑥
2
−
6.42
𝑥
+
2.98
)
+
(
2.98
−
6.42
𝑥
+
0.923
𝑥
2
)
=
3.903
𝑥
2
−
12.84
𝑥
+
3.903
left parenthesis 0.923 x squared minus 6.42 x plus 2.98 right parenthesis plus left parenthesis 2.98 minus 6.42 x plus 0.923 x squared right parenthesis equals 3.903 x squared minus 12.84 x plus 3.903

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=1.846x2−12.84x+5.96
(
0.923
𝑥
2
−
6.42
𝑥
+
2.98
)
+
(
2.98
−
6.42
𝑥
+
0.923
𝑥
2
)
=
1.846
𝑥
2
−
12.84
𝑥
+
5.96
left parenthesis 0.923 x squared minus 6.42 x plus 2.98 right parenthesis plus left parenthesis 2.98 minus 6.42 x plus 0.923 x squared right parenthesis equals 1.846 x squared minus 12.84 x plus 5.96

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=10.994x2+5.96
(
0.923
𝑥
2
−
6.42
𝑥
+
2.98
)
+
(
2.98
−
6.42
𝑥
+
0.923
𝑥
2
)
=
10.994
𝑥
2
+
5.96
left parenthesis 0.923 x squared minus 6.42 x plus 2.98 right parenthesis plus left parenthesis 2.98 minus 6.42 x plus 0.923 x squared right parenthesis equals 10.994 x squared plus 5.96

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=2x2−13x+6

Answer 2

Which expression is equivalent to 3x2y3z
3
𝑥
2
𝑦
3
𝑧
?(1 point)
Responses

x2y⋅3yz⋅y
𝑥
2
𝑦
⋅
3
𝑦
𝑧
⋅
𝑦
x squared y times 3 y z times y

x2y2z2⋅3y
𝑥
2
𝑦
2
𝑧
2
⋅
3
𝑦
x squared y squared z squared times 3 y

xy⋅3yz
𝑥
𝑦
⋅
3
𝑦
𝑧
x y times 3 y z

xy3⋅yz⋅3x

Answer 3

Multiply the polynomial expressions (2x2−3)(x+5)
(
2
𝑥
2
−
3
)
(
𝑥
+
5
)
.(1 point)
Responses

2x3+10x2−3x−15
2
𝑥
3
+
10
𝑥
2
−
3
𝑥
−
15
2 x cubed plus 10 x squared minus 3 x minus 15

2x2−3x−15
2
𝑥
2
−
3
𝑥
−
15
2 x squared minus 3 x minus 15

3x3+10x2−3x−15
3
𝑥
3
+
10
𝑥
2
−
3
𝑥
−
15
3 x cubed plus 10 x squared minus 3 x minus 15

3x2+7x−15

Answer 4

Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses

5⋅3x
5
⋅
3
𝑥
5 times 3 over x

(13)(x4−2)
(
1
3
)
(
𝑥
4
−
2
)
left parenthesis Start Fraction 1 over 3 End Fraction right parenthesis left parenthesis Start Fraction x over 4 End Fraction minus 2 right parenthesis

5x−1⋅2
5
𝑥
−
1
⋅
2

5 times x inverse times 2

2x3+x2

Answer 5

What must be true about this multiplication problem?

(9x3+76)(x−25)
(
9
𝑥
3
+
7
6
)
(
𝑥
−
2
5
)


(1 point)
Responses

The result is a polynomial.
The result is a polynomial.

The result is not a closed system.
The result is not a closed system.

The result is unknown.
The result is unknown.

The result is not a polynomial.

Answer 6

To find the sum of the given expression \((4p−2p^2+8)+(5p^2+2p)+(7−3p)\), we first need to combine all the like terms.

Let's break it down step by step:

1. **Grouping Similar Types**:
- Collect all \(p^2\) terms:
\(-2p^2 + 5p^2\)
- Collect all \(p\) terms:
\(4p + 2p - 3p\)
- Collect all constant terms:
\(8 + 7\)

2. **Combining the \(p^2\) Terms**:
- \(-2p^2 + 5p^2 = 3p^2\)

3. **Combining the \(p\) Terms**:
- \(4p + 2p - 3p = 3p\)

4. **Combining the Constant Terms**:
- \(8 + 7 = 15\)

5. **Putting it all Together**:
- Combine the results: \(3p^2 + 3p + 15\)

So, the final sum is:

\[
3p^2 + 3p + 15
\]

Thus, the correct response is:

**3p² + 3p + 15**.

Answer 7

To show that polynomials are a closed system under addition, we need to demonstrate that when we add two polynomials, the result is also a polynomial.

Let's analyze the provided options one by one:

1. **First option:**
\[
(0.923x^2−6.42x+2.98) + (2.98−6.42x+0.923x^2) = 3.903x^2−12.84x+3.903
\]
- When we add these two polynomials, we correctly combine like terms:
- \(0.923x^2 + 0.923x^2 = 1.846x^2\)
- \(-6.42x - 6.42x = -12.84x\)
- \(2.98 + 2.98 = 5.96\)

However, the result should be \(1.846x^2 - 12.84x + 5.96\), which does not match. **This option is incorrect.**

2. **Second option:**
\[
(0.923x^2−6.42x+2.98) + (2.98−6.42x+0.923x^2) = 1.846x^2−12.84x+5.96
\]
- Similar to the first option, the like terms are added correctly, and the result is correctly calculated. **This option is correct.**

3. **Third option:**
\[
(0.923x^2−6.42x+2.98) + (2.98−6.42x+0.923x^2) = 10.994x^2 + 5.96
\]
- The result does not match after combining the like terms. **This option is incorrect.**

4. **Fourth option:**
\[
(0.923x^2−6.42x+2.98) + (2.98−6.42x+0.923x^2) = 2x^2−13x+6
\]
- This result does not match and is incorrect after evaluating. **This option is incorrect.**

Therefore, the correct response that shows that polynomials are a closed system under addition is:

**(0.923x²−6.42x+2.98) + (2.98−6.42x+0.923x²) = 1.846x²−12.84x+5.96**.

Answer 8

To determine which expression is equivalent to \(3x^2y^3z\), let's analyze each of the provided options step by step:

1. **First option:**
\[
x^2y \cdot 3yz \cdot y
\]
- Simplifying this:
- \(x^2y \cdot 3yz \cdot y = 3x^2y^2z \cdot y = 3x^2y^3z\)
- This option **is equivalent to \(3x^2y^3z\)**.

2. **Second option:**
\[
x^2y^2z^2 \cdot 3y
\]
- Simplifying this:
- \(x^2y^2z^2 \cdot 3y = 3x^2y^3z^2\)
- This option **is NOT equivalent to \(3x^2y^3z\)**.

3. **Third option:**
\[
xy \cdot 3yz
\]
- Simplifying this:
- \(xy \cdot 3yz = 3xy^2z\)
- This option **is NOT equivalent to \(3x^2y^3z\)**.

4. **Fourth option:**
\[
xy^3 \cdot yz \cdot 3x
\]
- Simplifying this:
- \(xy^3 \cdot yz \cdot 3x = 3x^2y^4z\)
- This option **is NOT equivalent to \(3x^2y^3z\)**.

Based on the analyses above, the correct response is:

**First option:** \(x^2y \cdot 3yz \cdot y\).

Answer 9

To multiply the polynomial expressions \( (2x^2 - 3)(x + 5) \), we will use the distributive property (also known as the FOIL method for binomials):

### Step-by-Step Multiplication

1. **Distribute \(2x^2\) to both terms in \((x + 5)\)**:
\[
2x^2 \cdot x = 2x^3
\]
\[
2x^2 \cdot 5 = 10x^2
\]

2. **Distribute \(-3\) to both terms in \((x + 5)\)**:
\[
-3 \cdot x = -3x
\]
\[
-3 \cdot 5 = -15
\]

3. **Combine all the products together**:
\[
2x^3 + 10x^2 - 3x - 15
\]

### Final Result
The result of multiplying the polynomials \( (2x^2 - 3)(x + 5) \) is:
\[
2x^3 + 10x^2 - 3x - 15
\]

### Conclusion
Based on the given options, the correct response is:

**2x³ + 10x² - 3x - 15.**

Answer 10

To determine which option demonstrates that polynomials form a closed system under multiplication, we need to ensure that when two polynomials are multiplied, the result is also a polynomial.

Let's analyze each option:

1. **First option:**
\[
5 \cdot 3x
\]
- This simplifies to \(15x\), which is a polynomial. **This option demonstrates closure under multiplication.**

2. **Second option:**
\[
(13)(x^4 - 2)
\]
- This simplifies to \(13x^4 - 26\), which is also a polynomial. **This option demonstrates closure under multiplication.**

3. **Third option:**
\[
5x^{-1} \cdot 2
\]
- Here, \(x^{-1}\) indicates that this term is not a polynomial (since it includes a negative exponent). Thus, the product \(10x^{-1}\) is not a polynomial. **This option does not demonstrate closure under multiplication.**

4. **Fourth option:**
\[
2x^3 + x^2
\]
- This is a polynomial expression, but it does not show multiplication of two polynomials. **This option does not demonstrate closure under multiplication.**

### Conclusion
The first two options demonstrate that polynomials form a closed system under multiplication:

- **\(5 \cdot 3x\)**
- **\((13)(x^4 - 2)\)**

Both of these options confirm that the result after multiplying polynomials remains a polynomial.

If you need to select just one response that clearly shows this property, either the first option or the second option would be valid choices.

Answer 11

To analyze the multiplication problem \((9x^3 + 76)(x - 25)\), we need to determine what is true about this multiplication.

### Analysis:

1. **Result of the Multiplication**:
- When you multiply two polynomials (in this case, \(9x^3 + 76\) and \(x - 25\)), the result will also be a polynomial because polynomials are closed under multiplication.

2. **Polynomials Definition**:
- A polynomial is an expression that consists of variables raised to non-negative integer powers, along with coefficients that are real (or complex) numbers. Both \(9x^3 + 76\) and \(x - 25\) are polynomials.

3. **Multiplication of Polynomials**:
- Performing the multiplication:
\[
(9x^3 + 76)(x - 25) = 9x^3 \cdot x + 9x^3 \cdot (-25) + 76 \cdot x + 76 \cdot (-25)
\]
- This will yield terms of the form \(9x^4 - 225x^3 + 76x - 1900\), which is also a polynomial.

### Conclusion:

Given the analysis, the only true statement about the multiplication problem is:

**The result is a polynomial.**

So, the correct response is:

**The result is a polynomial.**