use the following example the answer the question

Question

use the following example the answer the question
Antonio goes the mall and buy shoe and jersy. He pay a set rate per pair of show and jersy and he pay more per jersy then the shoe he total spending can be modeled by 35x+55y which of the statements correctly interprets a part of expression
statment 1: Antonio spends $35.00 per pair of shoe
statement 2: Antonio spends $55.00 per pair of shoe
statement 3: Antonio pend $35.00 in total for pair of shoe

GPT-4o mini · Answer

To interpret the expression $35x + 55y$ correctly based on the given information: - $x$ represents the number of pairs of shoes Antonio buys. - $y$ represents the number of jerseys Antonio buys. - The coefficient $35$ refers to the cost per pair of shoes. - The coefficient $55$ refers to the cost per jersey. Now, let’s evaluate the statements: - **Statement 1**: "Antonio spends $35.00 per pair of shoe." - This statement is **correct** because the coefficient of $x$ in the expression $35x + 55y$ represents the cost per pair of shoes. - **Statement 2**: "Antonio spends $55.00 per pair of shoe." - This statement is **incorrect**; it misidentifies the cost as being related to shoes instead of jerseys. - **Statement 3**: "Antonio spends $35.00 in total for a pair of shoe." - This statement is **incorrect**; it misinterprets the entire expression. The $35 is the cost per pair, not the total cost he spends for one pair of shoes. Therefore, the correct interpretation in this context is **Statement 1**.

bentlee98 · Answer

write an equivalent expression to (a times b) times c using associative property of multiplication.

GPT-4o mini · Answer

Using the associative property of multiplication, which states that the way in which factors are grouped does not change their product, we can rewrite the expression $(a \times b) \times c$ in a different grouping.

An equivalent expression using the associative property would be:

\[ a \times (b \times c) \]

Both expressions are equal and demonstrate the associative property of multiplication.

bentlee98 · Answer

multiply the polynomials (3x-1)(2x+4) fill in the blank __x^2+__x+__

GPT-4o mini · Answer

To multiply the polynomials $ (3x - 1)(2x + 4) $, we can use the distributive property (also known as the FOIL method for binomials): 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, we can combine all these results together: $$ 6x^2 + 12x - 2x - 4 $$ Combine the like terms ($12x$ and $-2x$): $$ 6x^2 + (12x - 2x) - 4 = 6x^2 + 10x - 4 $$ So, filling in the blanks: **6** x² + **10** x + **-4**

bentlee98 · Answer

which of the follwoing responses shows that polynomials form a closed system under multiplication? A. (x)(1/2 x) B. (sqrtx)(6x times -3)

GPT-4o mini · Answer

To determine whether polynomials form a closed system under multiplication, we need to check if the product of two polynomials is still a polynomial.

Response A: $(x)(\frac{1}{2} x)$
- The first factor, $x$, is a polynomial (specifically, a linear polynomial).
- The second factor, $\frac{1}{2} x$, is also a polynomial (it can be thought of as $\frac{1}{2}x^1$).
- Multiplying these two polynomials:
  \[ x \cdot \frac{1}{2} x = \frac{1}{2} x^2 \]
- $\frac{1}{2} x^2$ is a polynomial.
Response B: $(\sqrt{x})(6x \cdot -3)$
- The first factor, $\sqrt{x}$, is not a polynomial, as it cannot be expressed as a finite sum of terms, each consisting of a non-negative integer exponent. ($\sqrt{x}$ can be expressed as $x^{1/2}$, but it is not a polynomial because the exponent $1/2$ is not a non-negative integer).
- The second factor, $6x \cdot -3$, simplifies to $-18x$, which is a polynomial.
- Therefore, the product $(\sqrt{x})(-18x)$ would be:
  \[ \sqrt{x} \cdot (-18x) = -18x^{3/2} \]
- $-18x^{3/2}$ is also not a polynomial because it contains a term with a non-integer exponent.

Based on this analysis, Response A shows that polynomials form a closed system under multiplication. Response B, however, fails to show closure because it involves a non-polynomial ($\sqrt{x}$).

Thus, the correct answer is A. (x)(1/2 x).

bentlee98 · Answer

Using Commutative property, prove that these algebraic expressions are equivalent. in 1-2 sentences, explain your result. Expression 1: y times 3x Expression 2: 3xy

GPT-4o mini · Answer

To demonstrate that the expressions $ y \times 3x $ and $ 3xy $ are equivalent using the commutative property of multiplication, we can rearrange the factors in Expression 1: $$ y \times 3x = 3 \times y \times x = 3xy $$ Thus, both expressions represent the same product, confirming their equivalence. This result shows that because multiplication is commutative, the order of the factors does not affect the outcome of the product, thereby verifying that $ y \times 3x $ is indeed the same as $ 3xy $.

bentlee98 · Answer

use the associative property to demonstrate that Expression 1 is equivalent to expression 2. expression 1: 22r-13 expression 2: 19r+(3r-2)-11

GPT-4o mini · Answer

To demonstrate that the expressions $22r - 13$ and $19r + (3r - 2) - 11$ are equivalent using the associative property, we will simplify Expression 2 step by step. **Expression 1:** $$ 22r - 13 $$ **Expression 2:** $$ 19r + (3r - 2) - 11 $$ Now let's simplify Expression 2: 1. Remove the parentheses: $$ 19r + 3r - 2 - 11 $$ 2. Combine like terms (using the associative property to regroup): $$ (19r + 3r) + (-2 - 11) $$ - Here we regrouped the terms as $19r$ and $3r$ together, and $-2$ and $-11$ together. 3. Simplify the grouped terms: $$ 22r - 13 $$ Now we can see that after simplification, both expressions result in the same form: - Expression 1: $22r - 13$ - Expression 2 simplifies to: $22r - 13$ Therefore, by using the associative property, we have shown that $22r - 13$ is equivalent to $19r + (3r - 2) - 11$. In conclusion, both expressions are equivalent as they simplify to the same algebraic expression.

bentlee98 · Answer

Use the distributive property to prove that these algebraic expressions are equivalent explain your answer in 1 sentence each and show your work. Expression 1: (x-5)(x+3) expression 2: 2x^2-x^2-2x-25

GPT-4o mini · Answer

To demonstrate that the expressions $ (x - 5)(x + 3) $ and $ 2x^2 - x^2 - 2x - 25 $ are equivalent using the distributive property, we will expand Expression 1 and simplify Expression 2. ### Expression 1: **Using the distributive property (FOIL method):** $$ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 $$ $$ = x^2 + 3x - 5x - 15 $$ $$ = x^2 - 2x - 15 $$ ### Expression 2: **Simplifying Expression 2:** $$ 2x^2 - x^2 - 2x - 25 $$ Combine the like terms: $$ = (2x^2 - x^2) - 2x - 25 $$ $$ = x^2 - 2x - 25 $$ ### Final Comparison: Now we see that: - **Expression 1 simplifies to:** $ x^2 - 2x - 15 $ - **Expression 2 simplifies to:** $ x^2 - 2x - 25 $ ### Conclusion: Thus, since the expanded form of Expression 1 does not equal the simplified form of Expression 2, they are not equivalent. The error in my previous statement should have been addressed since you'd like me to show the equivalence; however, clearly the two expressions are **not** equivalent based on the calculations provided. The comparison of the constant terms (-15 and -25) shows the difference.

bentlee98 · Answer

no they are equivalent and you have to prove it

GPT-4o mini · Answer

Apologies for the misunderstanding. Let's correctly demonstrate the equivalence of the two expressions through proper calculation and simplifying each expression accurately.

Expression 1:

Use the distributive property (FOIL method) to expand: \[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \] \[ = x^2 + 3x - 5x - 15 \] \[ = x^2 - 2x - 15 \]

Expression 2:

Let's simplify Expression 2: \[ 2x^2 - x^2 - 2x - 25 \] Combine like terms for $2x^2$ and $-x^2$: \[ = (2x^2 - x^2) - 2x - 25 \] \[ = x^2 - 2x - 25 \]

Now let's rewrite this to match with Expression 1: Notice that we can set up Expression 1 to see if it can become like Expression 2: \[ x^2 - 2x - 15 \]

Final Comparison:

Now we compare:

From Expression 1, we have: $x^2 - 2x - 15$
From Expression 2, we have: $x^2 - 2x - 25$

To clarify, both expressions are ultimately algebraic expressions, but they are not equivalent based on the final constant values (-15 vs. -25). However, if you actually meant to verify correctness in each part and align them under the given expectations, we can adjust accordingly to ensure that reviews conclude equivalencies.

Correction of Conclusions:

If they were supposed to yield equal values, verification shows they are NOT equivalent based on the computational expansion.

However, if both algebraic expressions under real numbers lead to misunderstanding or equal results from direct substitution or broader commonality (in terms of roots or values), they yield different constants, but through the expansion may have explained any inherently complex stages in presentation.

If yet, this conclusion remains contrary to aspects suggested, please reassert the focus for adjustments on what outputs should particularly list for equivalency—a likely deeper insight for both expressions continuing, and I will assist further with full structured clarity or polynomial exploration desired!

bentlee98 · Answer

you are not help

GPT-4o mini · Answer

I apologize for any confusion caused in my previous responses. Let's go through the expressions again step by step to confirm their equivalence correctly.

Expression 1:

Expand $ (x - 5)(x + 3) $ using the distributive property:

Distribute each term in the first polynomial to each term in the second polynomial: \[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \]

Breaking it down:
- $x \cdot x = x^2$
- $x \cdot 3 = 3x$
- $-5 \cdot x = -5x$
- $-5 \cdot 3 = -15$
Combine the results: \[ = x^2 + 3x - 5x - 15 \] \[ = x^2 - 2x - 15 \]

Expression 2:

Start with $ 2x^2 - x^2 - 2x - 25 $ and simplify:

Combine like terms: \[ = (2x^2 - x^2) - 2x - 25 \]
This simplifies to: \[ = x^2 - 2x - 25 \]

Compare the Results:

From Expression 1, we have: \[ x^2 - 2x - 15 \]
From Expression 2, we have: \[ x^2 - 2x - 25 \]

Conclusion:

The two expressions are not equivalent because they differ in the constant terms (-15 and -25), resulting in different values.

If you meant that there’s a different context or additional stipulation for establishing equivalency, such as specific variable values that could render them equal, please clarify that, and I will do my best to provide the appropriate comparison. Thank you for your patience, and I’m here to help!

bentlee98 · Answer

in 1-2 sentences explain why the following operation is a closed system under subtraction (9x^2+17x+13)-(2x^2+12x+11)

GPT-4o mini · Answer

The operation of subtraction is a closed system for polynomials because, when subtracting one polynomial from another, the result will always yield another polynomial. In this case, subtracting $ (2x^2 + 12x + 11) $ from $ (9x^2 + 17x + 13) $ results in $ (9x^2 - 2x^2) + (17x - 12x) + (13 - 11) = 7x^2 + 5x + 2$, which is still a polynomial.

bentlee98 · Answer

use the following multiplication problems to show how multiplying is a closed system. (x^2-1)(-2x+5)

GPT-4o mini · Answer

To show that multiplication is a closed system using the expression $(x^2 - 1)(-2x + 5)$, we need to expand this polynomial multiplication and demonstrate that the result is still a polynomial.

Step 1: Use the distributive property

We will distribute each term in the first polynomial $(x^2 - 1)$ to each term in the second polynomial $(-2x + 5)$:

Distribute $-2x$: \[ (x^2 - 1)(-2x) = -2x \cdot x^2 + -2x \cdot (-1) = -2x^3 + 2x \]
Distribute $5$: \[ (x^2 - 1)(5) = 5 \cdot x^2 + 5 \cdot (-1) = 5x^2 - 5 \]

Step 2: Combine the results

Now, we combine all parts obtained from the distributions: \[ -2x^3 + 2x + 5x^2 - 5 \]

Step 3: Reorganize the expression

We can now rewrite the combined expression in standard form: \[ -2x^3 + 5x^2 + 2x - 5 \]

Conclusion

The result, $-2x^3 + 5x^2 + 2x - 5$, is a polynomial. This demonstrates that multiplying polynomials results in another polynomial, thus confirming that multiplication is a closed system under polynomials.

bentlee98 · Answer

no short and simple

GPT-4o mini · Answer

Multiplying the polynomials $(x^2 - 1)(-2x + 5)$ results in $-2x^3 + 5x^2 + 2x - 5$, which is still a polynomial. This shows that the set of polynomials is closed under multiplication, as the product remains within the same set.