Question

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

3b2+18b−48
3 b squared plus 18 b minus 48

b2+18b−36
b squared plus 18 b minus 36

b2+30b−48
b squared plus 30 b minus 48

3b2+30b−48

Answer 1

2 of 52 of 5 Items
Question
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

x3y−x2y2+2x2−2y
x cubed y minus x squared y squared plus 2 x squared minus 2 y

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

x4y−x2y2+2x2−2y

Answer 2

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

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

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

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

2x3+x3y5

Answer 3

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

x2+2x+1
x squared plus 2 x plus 1

(x22)(1x)

Answer 4

(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

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

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

x2y3+5x4−8y3−15x2

Answer 5

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials. (1/2x2−3)(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 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 x2−−√y3+4x4−15x2 Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 15 x squared x2y3+5x4−8y3−15x2

Answer 6

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

1. First, we multiply the first terms:
\[
b \cdot 3b = 3b^2
\]

2. Next, we multiply the outer terms:
\[
b \cdot (-6) = -6b
\]

3. Then we multiply the inner terms:
\[
8 \cdot 3b = 24b
\]

4. Finally, we multiply the last terms:
\[
8 \cdot (-6) = -48
\]

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

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

So, the simplified expression is:
\[
\boxed{3b^2 + 18b - 48}
\]

Answer 7

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

1. **Distributing \(x^2y\)**:
- \(x^2y \cdot x^2 = x^{2+2}y = x^4y\)
- \(x^2y \cdot (-y) = -x^2y^2\)

2. **Distributing \(2\)**:
- \(2 \cdot x^2 = 2x^2\)
- \(2 \cdot (-y) = -2y\)

Now, combining all the terms together, we have:

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

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

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

Answer 8

To find the expression equivalent to \(x^3(2 + y^5)\), we will distribute \(x^3\) to each term inside the parentheses.

1. Distributing \(x^3\):
- \(x^3 \cdot 2 = 2x^3\)
- \(x^3 \cdot y^5 = x^3y^5\)

So, when combining these results, we get:

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

Thus, the expression equivalent to \(x^3(2 + y^5)\) is:

\[
\boxed{2x^3 + x^3y^5}
\]

Answer 9

To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of two polynomials is also a polynomial.

Let’s analyze the provided options:

1. **\((x−−√)(x+1)\)**: This involves the square root of \(x\), which is not a polynomial. This expression represents an algebraic expression but not a polynomial in the traditional sense.

2. **\((x^2 + 1)(x - \frac{1}{2})\)**: This is the product of two polynomials \((x^2 + 1)\) and \((x - \frac{1}{2})\). The result will be a polynomial, demonstrating closure.

3. **\(x^2 + 2x + 1\)**: This expression is already in polynomial form but does not show that it results from a multiplication of polynomials.

4. **\((x^{22})(1x)\)**: This is the multiplication of two polynomials \(x^{22}\) and \(x\), which will also yield a polynomial, demonstrating closure.

Among these options, the best responses that clearly demonstrate that polynomials form a closed system under multiplication are:

- **\((x^2 + 1)(x - \frac{1}{2})\)**
- **\((x^{22})(1x)\)**

Since you may be required to choose only one response, **\((x^2 + 1)(x - \frac{1}{2})\)** is the clearer demonstration of closure under multiplication, as it is more straightforwardly illustrative of polynomial multiplication.

So, the right response demonstrating closure under multiplication is:

\[
\boxed{(x^2 + 1)(x - \frac{1}{2})}
\]

Answer 10

To multiply the polynomials \((12x^2 - 3)(4y^3 + 5x^2)\), we'll use the distributive property (also known as the FOIL method for binomials).

1. First, we will distribute \(12x^2\) and then \(-3\) across the terms in the second polynomial.

**Distributing \(12x^2\)**:
- \(12x^2 \cdot 4y^3 = 48x^2y^3\)
- \(12x^2 \cdot 5x^2 = 60x^4\)

**Distributing \(-3\)**:
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)

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

Next, we can rearrange the terms in standard polynomial form:
\[
60x^4 + 48x^2y^3 - 15x^2 - 12y^3
\]

Now, let's look at the provided responses and identify the one that matches our result:
1. **\(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\)**: Incorrect.
2. **\(\frac{5}{2}x^2y^3 + 2x^4 - 8y^3 + 15x^2\)**: Incorrect.
3. **\(\sqrt{x^2}y^3 + 4x^4 - 15x^2\)**: Incorrect.
4. **\(x^2y^3 + 5x^4 - 8y^3 - 15x^2\)**: Incorrect.

It seems that none of the provided answers directly match the result \(60x^4 + 48x^2y^3 - 15x^2 - 12y^3\).

However, if I inspect closely, if one of the answers should have included \(48x^2y^3\) and \(60x^4\) but it's not in your options.

Let's try to summarize or just match:

The closest or most logical thing that could come from the multiplication of the two polynomials would generally look like the expansions we did. If there's no exact match, we may need to conclude that the given options might not reflect the output correctly.

Thus, based on our multiplication, our result remains as:
\[
\text{Final result: } 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \text{ (not in your choices)}
\]

If there's a misunderstanding about the inputs of the options, please clarify.

Answer 11

To find the product of the polynomials \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\), we'll use the distributive property:

1. **Distributing \(\frac{1}{2}x^2\)**:
- \(\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3\)
- \(\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4\)

2. **Distributing \(-3\)**:
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)

Now, we will combine these results:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

Now, let's rearrange the terms in a standard polynomial form:
\[
\frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3
\]

Now, let's evaluate the provided responses:

1. **\(2x^2y^3 + 52x^4 - 12y^3 - 15x^2\)**: Incorrect, the coefficient of \(x^4\) should be \(\frac{5}{2}\), not \(52\).

2. **\(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\)**: This matches! This is exactly what we found.

3. **\(52x^2y^3 + 2x^4 - 8y^3 + 15x^2\)**: Incorrect.

4. **\(\frac{5}{2}x^2y^3 + 2x^4 - 8y^3 + 15x^2\)**: Incorrect.

5. **\(x^2\sqrt{y^3} + 4x^4 - 15x^2\)**: Incorrect, as this involves a square root which is not polynomial.

6. **\(x^2y^3 + 5x^4 - 8y^3 - 15x^2\)**: Incorrect.

The correct response demonstrating the closure property of multiplication for polynomials is:

\[
\boxed{2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2}
\]