Question

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

(x)(1/2x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis

(x−−√)(6x ⋅−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren

3x2/4−1
Start Fraction 3 x squared over 4 minus 1 End Fraction

1/4x3⋅5

Answer 1

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


(2 points)

Answer 2

To determine which response demonstrates that polynomials form a closed system under multiplication, we need to identify a response where the multiplication of polynomials results in another polynomial.

Let's evaluate the provided options:

1. **(x)(1/2x):**
- This is the multiplication of two polynomials: \(x\) and \(\frac{1}{2}x\). The result is \(\frac{1}{2}x^2\), which is a polynomial.

2. **(√x)(6x ⋅−3):**
- Here we have \(\sqrt{x}\) (which is not a polynomial for non-integer powers) and \(6x \cdot -3\). The product involves \(\sqrt{x}\), making the result not a polynomial.

3. **3x²/4−1:**
- This expression is a polynomial itself (in standard form, it's a polynomial despite having a fractional coefficient and a constant). However, it’s not a result of multiplication explicitly defined.

4. **1/4x³⋅5:**
- This multiplication involves \(\frac{1}{4}x^3\) (which is a polynomial) and \(5\) (a constant polynomial). The result is \(\frac{5}{4}x^3\), which is also a polynomial.

Based on this evaluation, the responses that show polynomial multiplication resulting in another polynomial are:

- **(x)(1/2x)**
- **1/4x³⋅5**

However, if we're asked to show that polynomials form a closed system specifically under multiplication, the first response **(x)(1/2x)** is the clearest example since both factors are simple polynomials, and their product directly demonstrates closure.

Answer 3

To prove that the expressions \( y \cdot 3x \) and \( 3xy \) are equivalent using the Commutative Property of multiplication, we can rearrange the factors in Expression 1 as follows:

Starting with Expression 1:
\[ y \cdot 3x = 3 \cdot y \cdot x \]

By the Commutative Property, we can change the order of multiplication:
\[ 3 \cdot y \cdot x = 3xy \]

This shows that \( y \cdot 3x \) is equivalent to \( 3xy \).

In conclusion, both expressions represent the same product, demonstrating that multiplication is commutative, enabling the rearrangement of factors.