Asked by Yes.

Add the polynomial expressions 5z−2z^2−4z+5 and 6z^2+3−7z
Responses:
4z2−6z+8
11z2+z−2
9z2−11z+8
4z2−z+8
Answered by Yes.
Which of the following correctly simplifies (9x^2+17x+13)−(2x^2+12x+11)
Responses:
12x2+2
7x2+29x+24
9x2+17x+13−2x2−12x−11
7x2+5x+2
Answered by Yes.
Multiply the polynomials (3x−1)(2x+4)
Answered by Yes.
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses:
1/4x^3⋅5
3x^2/4−1
(x)(1/2x)
(x−−√)(6x ⋅−3 )
Answered by Yes.
Using the Commutative Property, prove that these algebraic expressions are equivalent. In 1–2 sentences, explain your result.
Expression 1: y⋅3x
Expression 2: 3xy
Answered by Yes.
Use the Associative Property to demonstrate that Expression 1 is equivalent to Expression 2.
Expression 1: 22r−13
Expression 2: 19r+(3r−2)−11
Answered by GPT-4o mini
To add the polynomial expressions \(5z - 2z^2 - 4z + 5\) and \(6z^2 + 3 - 7z\), we start by combining like terms.

1. **Combine the first polynomial:**
\[
5z - 4z - 2z^2 + 5 = (5z - 4z) + (-2z^2) + 5 = 1z - 2z^2 + 5 = -2z^2 + z + 5
\]

2. **Write the second polynomial as is:**
\[
6z^2 - 7z + 3
\]

3. **Now, we add the two results:**
\[
(-2z^2 + z + 5) + (6z^2 - 7z + 3)
\]
Combine like terms:
- For \(z^2\) terms: \(-2z^2 + 6z^2 = 4z^2\)
- For \(z\) terms: \(z - 7z = -6z\)
- For constant terms: \(5 + 3 = 8\)

Putting it all together, we get:
\[
4z^2 - 6z + 8
\]

Thus, the correct response is:
**4z² - 6z + 8**.
Answered by GPT-4o mini
To simplify \((9x^2 + 17x + 13) - (2x^2 + 12x + 11)\), you need to distribute the negative sign across the second polynomial and then combine like terms.

1. **Distribute the negative sign:**
\[
(9x^2 + 17x + 13) - (2x^2 + 12x + 11) = 9x^2 + 17x + 13 - 2x^2 - 12x - 11
\]

2. **Combine like terms:**
- For \(x^2\) terms: \(9x^2 - 2x^2 = 7x^2\)
- For \(x\) terms: \(17x - 12x = 5x\)
- For constant terms: \(13 - 11 = 2\)

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

Thus, the correct response is:
**7x² + 5x + 2.**
Answered by GPT-4o mini
To multiply the polynomials \((3x - 1)(2x + 4)\), we can use the distributive property (also known as the FOIL method for binomials). Here are the steps:

1. **Multiply the first terms:**
\[
3x \cdot 2x = 6x^2
\]

2. **Multiply the outer terms:**
\[
3x \cdot 4 = 12x
\]

3. **Multiply the inner terms:**
\[
-1 \cdot 2x = -2x
\]

4. **Multiply the last terms:**
\[
-1 \cdot 4 = -4
\]

Now, combine all these results:
\[
6x^2 + 12x - 2x - 4
\]

Combine the like terms (\(12x - 2x\)):
\[
6x^2 + 10x - 4
\]

Thus, the product of the polynomials \((3x - 1)(2x + 4)\) is:
\[
\boxed{6x^2 + 10x - 4}
\]
Answered by GPT-4o mini
To show that polynomials form a closed system under multiplication, we need to demonstrate that the product of any two polynomials is also a polynomial.

Let's analyze each of the response options:

1. **\( \frac{1}{4}x^3 \cdot 5 \)**
This results in \( \frac{5}{4}x^3 \), which is a polynomial.

2. **\( \frac{3x^2}{4} - 1 \)**
This expression is a polynomial expression, but it does not perform a multiplication between two polynomials.

3. **\( (x)(\frac{1}{2}x) \)**
This results in \( \frac{1}{2}x^2 \), which is a polynomial.

4. **\( (\sqrt{x})(6x \cdot -3) \)**
The term \(\sqrt{x}\) is not a polynomial because polynomials cannot have fractional or irrational exponents. Thus, this expression does not show closure under multiplication.

The correct responses that demonstrate polynomials forming a closed system under multiplication are:
- **\( \frac{1}{4}x^3 \cdot 5 \)**
- **\( (x)(\frac{1}{2}x) \)**

However, if you are looking for a single response, then **\( \frac{1}{4}x^3 \cdot 5 \)** and **\( (x)(\frac{1}{2}x) \)** both are valid examples showing closure under multiplication. If only one option is to be chosen, either would suffice, but typically, \( \frac{1}{4}x^3 \cdot 5 \) is more straightforward.
Answered by GPT-4o mini
To prove that the expressions \( y \cdot 3x \) and \( 3xy \) are equivalent using the Commutative Property, we can rearrange the factors in \( y \cdot 3x \):

\[
y \cdot 3x = 3 \cdot y \cdot x = 3xy
\]

By the Commutative Property of multiplication, the order of the factors in a product does not change the result. Thus, the expressions \( y \cdot 3x \) and \( 3xy \) represent the same quantity, proving their equivalence.
Answered by GPT-4o mini
To demonstrate that the expressions \(22r - 13\) and \(19r + (3r - 2) - 11\) are equivalent using the Associative Property, we start by simplifying Expression 2.

### Expression 2:
\[
19r + (3r - 2) - 11
\]

First, we can simplify the expression inside the parentheses:
\[
= 19r + 3r - 2 - 11
\]

Next, combine the like terms:
\[
19r + 3r - 2 - 11 = (19r + 3r) + (-2 - 11) = 22r - 13
\]

Now we have:
\[
19r + (3r - 2) - 11 = 22r - 13
\]

### Conclusion:
Since we have shown that \(19r + (3r - 2) - 11\) simplifies to \(22r - 13\), we conclude that both expressions are equivalent. The Associative Property allows us to regroup terms without changing the result, which we used to combine \(19r\) and \(3r\) effectively.