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What is the product of the polynomials (x2y+2)(x2−y) ?(1 point)

Responses

x4y−xy+x2y
x superscript 4 baseline y minus x y plus x squared y

x4y+x2−2y
x superscript 4 baseline y plus x squared minus 2 y

x4y−x2y2+2x2−2y
x superscript 4 baseline y minus x squared y squared plus 2 x squared minus 2 y

x3y−x2y2+2x2−2y

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
1 month ago

To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we can use the distributive property (also known as the FOIL method in the case of binomials).

Let's distribute each term from the first polynomial to each term in the second polynomial:

1. Multiply \(x^2y\) by \(x^2\):
\[
x^2y \cdot x^2 = x^4y
\]

2. Multiply \(x^2y\) by \(-y\):
\[
x^2y \cdot (-y) = -x^2y^2
\]

3. Multiply \(2\) by \(x^2\):
\[
2 \cdot x^2 = 2x^2
\]

4. Multiply \(2\) by \(-y\):
\[
2 \cdot (-y) = -2y
\]

Now, we can combine all of these results to get the final form of the product:

\[
x^4y - x^2y^2 + 2x^2 - 2y
\]

Thus, the product of the polynomials \((x^2y + 2)(x^2 - y)\) is:

\[
\boxed{x^4y - x^2y^2 + 2x^2 - 2y}
\]

From the given responses, the correct option is:

**x4y−x2y2+2x2−2y** (which matches with the last option listed).

anonymous anonymous answered
1 month ago

Which expression is equivalent to x3(2+y5) ?(1 point)

Responses

2x3+x3y5
2 x cubed plus x cubed y superscript 5 baseline

x3+x3+xy+xy+xy
x cubed plus x cubed plus x y plus x y plus x y

x3+2+y5
x cubed plus 2 plus y superscript 5 baseline

2x3+y5

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
1 month ago

To find an expression equivalent to \( x^3(2 + y^5) \), we can use the distributive property. This property states that \( a(b + c) = ab + ac \).

Applying this to our expression:

\[
x^3(2 + y^5) = x^3 \cdot 2 + x^3 \cdot y^5 = 2x^3 + x^3y^5
\]

So, the correct equivalent expression is:

\[
2x^3 + x^3y^5
\]

From the given options, the correct response is:

**2x3 + x3y5** (the first option listed).

anonymous anonymous answered
1 month ago

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

Responses

(x−−√)(x+1)
left parenthesis Start Root x End Root right parenthesis left parenthesis x plus 1 right parenthesis

(x2+1)(x−12)
left parenthesis x squared plus 1 right parenthesis left parenthesis x minus Start Fraction 1 over 2 End Fraction right parenthesis

(x22)(1x)
left parenthesis Start Fraction x squared over 2 End Fraction right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis

x2+2x+1

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
1 month ago

To demonstrate that polynomials form a closed system under multiplication, we need an expression that, when multiplied, results in another polynomial.

Let's analyze the given responses:

1. **\((\sqrt{x})(x + 1)\)**: This expression involves \(\sqrt{x}\), which is not a polynomial because polynomials cannot have fractional or negative exponents. The product will contain a non-polynomial term.

2. **\((x^2 + 1)(x - \frac{1}{2})\)**: Both \(x^2 + 1\) and \(x - \frac{1}{2}\) are polynomials. The product of these two polynomials will be another polynomial. Therefore, this demonstrates closure under multiplication.

3. **\(\left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right)\)**: The term \(\frac{1}{x}\) is not a polynomial (it contains a negative exponent when rewritten as \(x^{-1}\)). Thus, the product is not guaranteed to be a polynomial.

4. **\(x^2 + 2x + 1\)**: This is a polynomial, but as a standalone expression, it does not demonstrate the closure under multiplication as it does not include a multiplication operation.

The correct response that demonstrates polynomials form a closed system under multiplication is:

**\((x^2 + 1)(x - \frac{1}{2})\)**.

anonymous anonymous answered
1 month ago

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials.

(12x2−3)(4y3+5x2)

(1 point)
Responses

2x2y3+52x4−12y3−15x2
2 x squared y cubed plus Start Fraction 5 over 2 End Fraction x superscript 4 baseline minus 12 y cubed minus 15 x squared

x2−−√y3+4x4−15x2
Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 15 x squared

52x2y3+2x4−8y3+15x2
Start Fraction 5 over 2 End Fraction x squared y cubed plus 2 x superscript 4 baseline minus 8 y cubed plus 15 x squared

x2y3+5x4−8y3−15x2

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
1 month ago

To find the product of the polynomials \((12x^2 - 3)(4y^3 + 5x^2)\), we will use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

### Steps to Multiply:

1. Distribute \(12x^2\) to both terms in \((4y^3 + 5x^2)\):
- \(12x^2 \cdot 4y^3 = 48x^2y^3\)
- \(12x^2 \cdot 5x^2 = 60x^4\)

2. Distribute \(-3\) to both terms in \((4y^3 + 5x^2)\):
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)

### Combine All Terms:
Now, we combine all the terms we obtained:

\[
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
\]

### Rearranging:

Rearranging the terms gives:

\[
60x^4 + 48x^2y^3 - 15x^2 - 12y^3
\]

Now let's review the options given to identify which one matches with our derived expression:

1. **\(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\)**: Not correct.
2. **\(\sqrt{x^2}y^3 + 4x^4 - 15x^2\)**: Not correct.
3. **\(\frac{5}{2}x^2y^3 + 2x^4 - 8y^3 + 15x^2\)**: Not correct.
4. **\(x^2y^3 + 5x^4 - 8y^3 - 15x^2\)**: Not correct.

None of the provided options directly match the simplified product we calculated: \(60x^4 + 48x^2y^3 - 15x^2 - 12y^3\).

However, the expression can be rearranged or factored and matched with factors, but the proper derived product doesn't perfectly align with any of the responses provided.

For clarity, the final product derived is:
\[
60x^4 + 48x^2y^3 - 15x^2 - 12y^3
\]

If you have more options or specific constraints in mind, please clarify!