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

Question

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

Responses

3b^2+18b−48
3 b squared plus 18 b minus 48

3b^2+30b−48
3 b squared plus 30 b minus 48

b^2+18b−36
b squared plus 18 b minus 36

b^2+30b−48

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9 answers
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Answer 1

Bot GPT 4-o mini answered
1 month ago

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

1. Multiply the first terms:
\[
b \cdot 3b = 3b^2
\]

2. Multiply the outer terms:
\[
b \cdot -6 = -6b
\]

3. Multiply the inner terms:
\[
8 \cdot 3b = 24b
\]

4. Multiply the last terms:
\[
8 \cdot -6 = -48
\]

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

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

So the simplified form of \( (b+8)(3b-6) \) is:
\[
\boxed{3b^2 + 18b - 48}
\]

The correct answer is:
**3b^2 + 18b - 48**.

Answer 2

x answered
1 month ago

What is the product of the polynomials (x^2y+2)(x^2−y) ?(1 point)

Responses

x^4y−xy+x^2y
x superscript 4 baseline y minus x y plus x squared y

x^3y−x^2y^2+2x^2−2y
x cubed y minus x squared y squared plus 2 x squared minus 2 y

x^4y+x^2−2y
x superscript 4 baseline y plus x squared minus 2 y

x^4y−x^2y^2+2x^2−2y

Answer 3

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 (or the FOIL method).

1. Distribute \( x^2 y \) to both terms in the second polynomial:
\[
x^2 y \cdot x^2 = x^4 y
\]
\[
x^2 y \cdot -y = -x^2 y^2
\]

2. Distribute \( 2 \) to both terms in the second polynomial:
\[
2 \cdot x^2 = 2x^2
\]
\[
2 \cdot -y = -2y
\]

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

So the product of \( (x^2y + 2)(x^2 - y) \) is:
\[
\boxed{x^4 y - x^2 y^2 + 2x^2 - 2y}
\]

The correct answer is:
**x^4y - x^2y^2 + 2x^2 - 2y**.

Answer 4

x answered
1 month ago

Which expression is equivalent to x^3(2+y^5) ?(1 point)

Responses

2x^3+y^5
2 x cubed plus y superscript 5 baseline

2x^3+x^3y^5
2 x cubed plus x cubed y superscript 5 baseline

x^3+x^3+xy+xy+xy
x cubed plus x cubed plus x y plus x y plus x y

x^3+2+y^5

Answer 5

Bot GPT 4-o mini answered
1 month ago

To simplify the expression \( x^3(2+y^5) \), we can use the distributive property:

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

This results in:

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

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

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

The correct answer is:
**2x^3 + x^3y^5**.

Answer 6

x answered
1 month ago

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

Responses

x^2+2x+1
x squared plus 2 x plus 1

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

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

(x^2+1)(x−1/2)

Answer 7

Bot GPT 4-o mini answered
1 month ago

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

The response that correctly illustrates this is:

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

When you multiply two polynomials \( (x^2 + 1) \) and \( (x - \frac{1}{2}) \), you will get:

1. \( x^2 \cdot x = x^3 \)
2. \( x^2 \cdot -\frac{1}{2} = -\frac{1}{2}x^2 \)
3. \( 1 \cdot x = x \)
4. \( 1 \cdot -\frac{1}{2} = -\frac{1}{2} \)

Combining these terms will result in a polynomial:

\[
x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}
\]

Thus, this product is still a polynomial, demonstrating closure under multiplication.

The other options do not necessarily demonstrate closure or involve products that result in non-polynomial expressions (for example, involving roots or fractions that are not polynomials).

Therefore, the correct response is:

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

Answer 8

x answered
1 month ago

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

(1/2x^2−3)(4y^3+5x^2)

(1 point)
Responses

2x^2y^3+5/2x^4−12y^3−15x^2
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

√x^2y^3 + 4x^4−15x^2
Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 15 x squared

x^2y^3+5/x^4−8y^3−15x^2
x squared y cubed plus Start Fraction 5 over x superscript 4 baseline End Fraction minus 8 y cubed minus 15 x squared

5/2x^2y^3+2x^4−8y^3+15x^2

Answer 9

Bot GPT 4-o mini answered
1 month ago

To find the product of the polynomials \( ( \frac{1}{2}x^2 - 3 )( 4y^3 + 5x^2 ) \) using multiplication, we can use the distributive property (or FOIL method):

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 can combine all these results together:

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

Putting it all together, the final polynomial product is:

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

Thus, the response that demonstrates the closure property of multiplication of polynomials is:

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

This response confirms that the product of two polynomials yields another polynomial, demonstrating closure under multiplication.