Question

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

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

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

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

3b^2+18b−48

Answer 1

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

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

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

x^4y+x^2−2y

Answer 2

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

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

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

x^3+2+y^5

Answer 3

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

(x2/2)(1/x)
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
x squared plus 2 x plus 1

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

(x−−√)(x+1)

Answer 4

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

1. **First:** \(b \cdot 3b = 3b^2\)
2. **Outer:** \(b \cdot (-6) = -6b\)
3. **Inner:** \(8 \cdot 3b = 24b\)
4. **Last:** \(8 \cdot (-6) = -48\)

Now we combine these results:

\[
3b^2 - 6b + 24b - 48
\]

Combine the like terms \(-6b + 24b\):

\[
3b^2 + 18b - 48
\]

Therefore, the simplified form of the polynomial is:

\[
\boxed{3b^2 + 18b - 48}
\]

So the correct response is \(3b^2 + 18b - 48\).

Answer 5

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

Let's multiply each term in the first polynomial by each term in the second polynomial.

1. First, distribute \(x^2y\):
- \(x^2y \cdot x^2 = x^4y\)
- \(x^2y \cdot (-y) = -x^2y^2\)

2. Now distribute \(2\):
- \(2 \cdot x^2 = 2x^2\)
- \(2 \cdot (-y) = -2y\)

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

Thus, the product of the polynomials is:
\[
\boxed{x^4y - x^2y^2 + 2x^2 - 2y}
\]

So the correct response is \(x^4y - x^2y^2 + 2x^2 - 2y\).

Answer 6

To find the expression equivalent to \( x^3(2 + y^5) \), we can use the distributive property (also known as the distributive law of multiplication over addition).

Distributing \( x^3 \) across \( (2 + y^5) \), we have:

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

This simplifies to:

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

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

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

So the correct response is \( 2x^3 + x^3y^5 \).

Answer 7

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 given options:

1. **\((x^2/2)(1/x)\)**:
- This simplifies to \((x^2/2)(1/x) = x/2\), which is not a polynomial because it contains a term with a negative exponent (the \(1/x\) contributes a power of \(-1\)). Thus, this does not demonstrate closure.

2. **\(x^2 + 2x + 1\)**:
- This is just a polynomial by itself. It does not show multiplication.

3. **\((x^2 + 1)(x - \frac{1}{2})\)**:
- When you multiply these two polynomials together, you will get another polynomial (since multiplying polynomials results in a polynomial). This is a valid response that demonstrates closure.

4. **\((x - \sqrt{})(x + 1)\)**:
- Depending on what the terms inside the square root are, if they are not fully specified, we can't evaluate this properly. However, if this represents multiplying two expressions with square roots (not particularly polynomial terms), it may break polynomial form depending on the complexity of the square root.

The best answer that clearly shows that polynomials form a closed system under multiplication is:

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

This response demonstrates that multiplying two polynomials results in another polynomial, satisfying the condition for closure under multiplication.