Asked by Finally Some Answers

Multiply the polynomial expression (x+2)(x−4) .(2 points)

()x^2+() x+()
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Multiply to simplify the polynomial expression (r−1)(r2−2r+3) .(2 points)

()r3+()r2+()r+()
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Simplify the polynomial expression (xy−2)(x2+1) .(2 points)

()x^3y+()x^2+()xy+()
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Which of the following shows that polynomials form a closed system under multiplication? Option #1: 3(2x+5)=6x+15 Option #2: 8x+7=5x+3x+1+6 Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24 Option #4: 2x12⋅3=6x−−√ (1 point)
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Which of the following options correctly describes if the following polynomial forms a closed system under multiplication? −3(5x+4y)=−15x−12y Option #1: Yes, because the result of multiplying the polynomials is also a polynomial. Option #2: No, because the exponent of x is not a positive integer. Option #3: No, because multiplying the polynomials resulted in subtraction. (1 point)
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Wrong answer it was Option#2
Answered by GPT-4o mini
To multiply the polynomial expression \((x + 2)(x - 4)\), you can use the distributive property (also known as the FOIL method for binomials).

1. First, multiply the first terms:
\[
x \cdot x = x^2
\]

2. Next, multiply the outer terms:
\[
x \cdot (-4) = -4x
\]

3. Then, multiply the inner terms:
\[
2 \cdot x = 2x
\]

4. Finally, multiply the last terms:
\[
2 \cdot (-4) = -8
\]

Now, combine all these results:
\[
x^2 - 4x + 2x - 8
\]

Combine the like terms (\(-4x + 2x\)):
\[
x^2 - 2x - 8
\]

So the final answer is:
\[
x^2 - 2x - 8
\]

In the format you requested, it would be:
- \(1\) for the coefficient of \(x^2\),
- \(-2\) for the coefficient of \(x\),
- \(-8\) for the constant term.

Thus, the answer is:
\[
(x^2 + (-2)x + (-8))
\]
Answered by GPT-4o mini
To multiply and simplify the expression \((r - 1)(r^2 - 2r + 3)\), we will distribute each term in the first polynomial to every term in the second polynomial.

1. Distribute \(r\):
\[
r \cdot r^2 = r^3
\]
\[
r \cdot (-2r) = -2r^2
\]
\[
r \cdot 3 = 3r
\]

2. Distribute \(-1\):
\[
-1 \cdot r^2 = -r^2
\]
\[
-1 \cdot (-2r) = 2r
\]
\[
-1 \cdot 3 = -3
\]

Now, combine all these results:
\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]

Combine the like terms:
1. For \(r^2\):
\(-2r^2 - r^2 = -3r^2\)

2. For \(r\):
\(3r + 2r = 5r\)

So, the final result is:
\[
r^3 - 3r^2 + 5r - 3
\]

In the format you requested, it would be:
- \(1\) for the coefficient of \(r^3\),
- \(-3\) for the coefficient of \(r^2\),
- \(5\) for the coefficient of \(r\),
- \(-3\) for the constant term.

Thus, the answer is:
\[
(r^3 + (-3)r^2 + 5r + (-3))
\]
Answered by GPT-4o mini
To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will distribute each term in the first polynomial to every term in the second polynomial.

1. Distributing \(xy\):
\[
xy \cdot x^2 = x^3y
\]
\[
xy \cdot 1 = xy
\]

2. Distributing \(-2\):
\[
-2 \cdot x^2 = -2x^2
\]
\[
-2 \cdot 1 = -2
\]

Now, combine all these results:
\[
x^3y + xy - 2x^2 - 2
\]

Now, rearranging them in standard form (descending order of powers) gives:
\[
x^3y - 2x^2 + xy - 2
\]

In the format you requested, it would be:
- \(1\) for the coefficient of \(x^3y\),
- \(-2\) for the coefficient of \(x^2\),
- \(1\) for the coefficient of \(xy\),
- \(-2\) for the constant term.

Thus, the answer is:
\[
(x^3y + (-2)x^2 + 1xy + (-2))
\]
Answered by GPT-4o mini
To determine which option shows that polynomials form a closed system under multiplication, we need to verify that the product of two polynomials results in another polynomial.

Here’s an evaluation of the options provided:

- **Option #1:** \(3(2x + 5) = 6x + 15\)
This shows scalar multiplication of a polynomial \(2x + 5\), resulting in another polynomial. However, this does not demonstrate multiplication of two polynomials.

- **Option #2:** \(8x + 7 = 5x + 3x + 1 + 6\)
This expression simplifies to \(8x + 7 = 8x + 7\), demonstrating that the expression is equal but does not pertain to multiplication of polynomials.

- **Option #3:** \((2x^2 - 4)(3y + 6) = 6x^2y + 12x^2 - 12y - 24\)
This shows the multiplication of two polynomials, resulting in another polynomial. This clearly demonstrates closure under multiplication.

- **Option #4:** \(2x^{12} \cdot 3 = 6x^{12}\)
This shows scalar multiplication of a polynomial \(2x^{12}\), resulting in another polynomial. However, like Option #1, it does not demonstrate multiplication of two polynomials.

**Conclusion:** The option that correctly shows that polynomials form a closed system under multiplication is:
**Option #3: \((2x^2 - 4)(3y + 6) = 6x^2y + 12x^2 - 12y - 24\)**
Answered by GPT-4o mini
To determine if the expression \(-3(5x + 4y) = -15x - 12y\) demonstrates that polynomials form a closed system under multiplication, let's analyze the options provided:

1. **Option #1:** Yes, because the result of multiplying the polynomials is also a polynomial.
- This option is correct. \(-15x - 12y\) is a polynomial, and multiplying a polynomial by a scalar (in this case, \(-3\)) results in another polynomial. Therefore, the set of polynomials is closed under multiplication.

2. **Option #2:** No, because the exponent of \(x\) is not a positive integer.
- This option is incorrect. The exponent of \(x\) in the polynomial \(-15x - 12y\) is \(1\), which is indeed a positive integer.

3. **Option #3:** No, because multiplying the polynomials resulted in subtraction.
- This option is also incorrect. Closure under multiplication refers to the final result being a polynomial, regardless of whether subtraction is involved in the original expression. The result \(-15x - 12y\) is still a polynomial.

Given the analysis, the correct choice is:

**Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.**
Answered by GPT-4o mini
I apologize for the confusion. Let's clarify the concepts related to polynomials and the closure property in the context of the provided options.

### Understanding Polynomial Closure Under Multiplication

Polynomials are expressions composed of variables, coefficients, and non-negative integer exponents. The closure property indicates that when you perform an operation (in this case, multiplication) on elements of the set (polynomials), the result should also belong to that set.

### Evaluating the Options

1. **Option #1:** Yes, because the result of multiplying the polynomials is also a polynomial.
- This states that the result should be a polynomial in order to affirm closure. Here, the result \(-15x - 12y\) is indeed a polynomial.

2. **Option #2:** No, because the exponent of \(x\) is not a positive integer.
- This statement implies a misunderstanding—while the exponents of polynomials are indeed non-negative integers, the expression \(-15x - 12y\) has an exponent of 1 for \(x\) and is therefore valid as a polynomial.

3. **Option #3:** No, because multiplying the polynomials resulted in subtraction.
- This option misinterprets the operation; subtraction in itself does not negate polynomial status.

### Conclusion

The assertion in **Option #2** is incorrect because the exponent of \(x\) is 1, which is indeed a positive integer.

Thus, the correct answer regarding the closure of polynomials under multiplication would still point towards **Option #1** being correct.

If your point relates to a different context or interpretation of the problem, please clarify, and I would be happy to reassess the explanation!