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Multiply the polynomial (b+8)(3b−6) to simplify.(1 point)

Responses:
b2+30b−48
b2+18b−36
3b2+30b−48
3b2+18b−48

11 answers

To multiply the polynomial \( (b + 8)(3b - 6) \), we can use the distributive property (also known as the FOIL method for binomials):

  1. First: Multiply the first terms:
    \( b \times 3b = 3b^2 \)

  2. Outside: Multiply the outer terms:
    \( b \times -6 = -6b \)

  3. Inside: Multiply the inner terms:
    \( 8 \times 3b = 24b \)

  4. Last: Multiply the last terms:
    \( 8 \times -6 = -48 \)

Now, combine all these results together:
\[ 3b^2 - 6b + 24b - 48 \]

Combine the like terms (the \( -6b \) and \( 24b \)):
\[ 3b^2 + (24b - 6b) - 48 = 3b^2 + 18b - 48 \]

Thus, the simplified form of \( (b + 8)(3b - 6) \) is: \[ 3b^2 + 18b - 48 \]

The correct response is: 3b² + 18b − 48.

What is the product of the polynomials (x2y+2)(x2−y) ?(1 point)
Responses:
x4y+x2−2y
x4y−x2y2+2x2−2y
x3y−x2y2+2x2−2y
x4y−xy+x2y

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

We'll distribute each term in the first polynomial \( (x^2y + 2) \) to each term in the second polynomial \( (x^2 - y) \):

  1. Distributing \( x^2y \):

    • \( x^2y \cdot x^2 = x^4y \)
    • \( x^2y \cdot (-y) = -x^2y^2 \)

  2. Distributing \( 2 \):

    • \( 2 \cdot x^2 = 2x^2 \)
    • \( 2 \cdot (-y) = -2y \)

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

Thus, the product of the polynomials \( (x^2y + 2)(x^2 - y) \) is: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

The correct response is: x⁴y − x²y² + 2x² − 2y.

Which expression is equivalent to x3(2+y5) ?(1 point)
Responses:
2x3+y5
x3+2+y5
x3+x3+xy+xy+xy
2x3+x3y5

To determine which expression is equivalent to \( x^3(2 + y^5) \), we can use the distributive property to expand it:

  1. Distributing \( x^3 \) across the terms inside the parentheses:
    • \( x^3 \cdot 2 = 2x^3 \)
    • \( x^3 \cdot y^5 = x^3y^5 \)

So, we have: \[ x^3(2 + y^5) = 2x^3 + x^3y^5 \]

Now, let's check the provided response options:

  1. \( 2x^3 + y^5 \) — This is not equivalent because it doesn't include \( x^3y^5 \).
  2. \( x^3 + 2 + y^5 \) — This is not equivalent because it doesn't maintain the multiplication by \( x^3 \).
  3. \( x^3 + x^3 + xy + xy + xy \) — This is not equivalent and is not a correct expansion.
  4. \( 2x^3 + x^3y^5 \) — This is equivalent to what we found.

Thus, the equivalent expression is: 2x³ + x³y⁵.

Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point)
Responses:
(x−−√)(x+1)
x^2+2x+1
(x2+1)(x−1?2)
(x^2/2)(1/x)

To demonstrate that polynomials form a closed system under multiplication, we need an example where the product of two polynomial expressions results in another polynomial.

Let’s analyze each of the given responses:

  1. \( (x - \sqrt{})(x + 1) \): This results in a polynomial. The product can be simplified to: \[ x^2 + x - \sqrt{} \] (Note: This appears to be incomplete; if it's meant to be \( (x - \sqrt{y})(x + 1) \), it's still a polynomial.)

  2. \( x^2 + 2x + 1 \): This is simply a polynomial and does not involve multiplication.

  3. \( (x^2 + 1)(x - 1^2) \): This involves multiplication of two polynomials, which will yield another polynomial. You can multiply it out as: \[ x^2(x - 1) + 1(x - 1) = x^3 - x^2 + x - 1 \] This product is indeed a polynomial.

  4. \( \left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right) \): Here, we are multiplying a polynomial by a non-polynomial term \( \frac{1}{x}\), which results in \( \frac{x^2}{2x} = \frac{x}{2} \). Since this is not a polynomial (specifically, it's not defined for \( x = 0 \)), it shows that this example does not demonstrate closure in polynomials.

From this analysis, the responses that demonstrate closure under multiplication by resulting in polynomials are:

  • \( (x^2 + 1)(x - 1^2) \)

So, the preferred answer is: (x^2 + 1)(x - 1^2).

My Apologies The Answers are:
(x−−√)(x+1)
x^2+2x+1
(x2+1)(x−1/2)
(x^2/2)(1/x)

No problem! Let's analyze the corrected options to determine which demonstrates that polynomials form a closed system under multiplication.

A closed system under multiplication for polynomials means that when two polynomials are multiplied together, the result is also a polynomial.

Let's look at each option:

  1. \( (x - \sqrt{})(x + 1) \): This expression involves the term \(\sqrt{}\), which suggests that it might not be complete or correctly formatted. However, if it was intended to be \( (x - \sqrt{y})(x + 1) \), the product still results in another polynomial, assuming \(\sqrt{y}\) is just a constant. Generally, if it contains non-polynomial terms (like square roots that are not constants), it would not demonstrate closure.

  2. \( x^2 + 2x + 1 \): This is already a polynomial, but it does not demonstrate the closure property since it doesn't involve a multiplication operation.

  3. \( (x^2 + 1)(x - \frac{1}{2}) \): This expression involves the multiplication of two polynomials:

    • The product will be \( (x^2 + 1)(x - \frac{1}{2}) = x^3 - \frac{1}{2}x^2 + x - \frac{1}{2} \), which is also a polynomial. This does demonstrate closure.

  4. \( \left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right) \): This results in \(\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x}{2}\). While \(\frac{x}{2}\) itself is a linear expression, it’s not strictly a polynomial form because it’s multiplied by \(\frac{1}{x}\), which is not a polynomial. Thus, it does not demonstrate closure under multiplication.

Given this analysis, the correct option that illustrates the closure of polynomials under multiplication is:

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

This multiplication of two polynomials results in a polynomial, demonstrating that polynomials are closed under multiplication.

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials.
(12x2−3)(4y3+5x2)
Responses:
x^2y^3+5/x4−8y^3−15x^2
5/2x^2y^3+2x^4−8y^3+15x^2
2x^2y^3+5/2x^4−12y3−15x^2
x^2−−√y^3+4x^4−15x^2

To demonstrate the Closure Property of multiplication for polynomials using the product \( (12x^2 - 3)(4y^3 + 5x^2) \), we first need to multiply these two polynomials together.

Let's carry out the multiplication step by step:

  1. Distribute \( 12x^2 \) across \( (4y^3 + 5x^2) \):

    • \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
    • \( 12x^2 \cdot 5x^2 = 60x^4 \)

  2. Distribute \( -3 \) across \( (4y^3 + 5x^2) \):

    • \( -3 \cdot 4y^3 = -12y^3 \)
    • \( -3 \cdot 5x^2 = -15x^2 \)

Now, we combine all these results together: \[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]

Rearranging the terms, we get: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]

Now, let's compare our product \( 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \) with the given response options:

  1. \( x^2y^3 + \frac{5}{x^4} - 8y^3 - 15x^2 \): This option includes a term \(\frac{5}{x^4}\), which is not a polynomial term, and does not match.

  2. \( \frac{5}{2}x^2y^3 + 2x^4 - 8y^3 + 15x^2 \): This does not match our results, and the coefficients are incorrect.

  3. \( 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \): This option also does not match what we found.

  4. \( x^2y^3 + 4x^4 - 15x^2 \): This does not match what we found either.

None of the provided responses appear to properly reflect the correct product obtained through the multiplication.

Final Result:

The correct product is: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]

If you're looking for a specific option among those provided that demonstrates the closure property, it's necessary to verify if the original expressions (before multiplication) are both polynomials, which they are. Hence, their multiplication resulting in another polynomial (as we calculated) indeed signifies the closure property of polynomials under multiplication. Unfortunately, it seems the specific result isn't reflected adequately in the provided options.