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

Question

Responses

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

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

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

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

Answer 1

To simplify the expression \( x^3(2 + y^5) \), we can distribute \( x^3 \) across the terms inside the parentheses:

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

From the responses provided, the expression that matches our simplified form is:

2x³ + x³y⁵

Therefore, the correct response is:

2x^3 + x^3y^5

Answer 2

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

Answer 3

To demonstrate that polynomials form a closed system under multiplication, we need to show that when two polynomials are multiplied together, the result is also a polynomial.

Let’s analyze each response:

1. \((\sqrt{x})(x + 1)\): The square root of \(x\) is not a polynomial, so this product may not be a polynomial.

2. \((x^2 + 1)(x - \frac{1}{2})\): Both \(x^2 + 1\) and \(x - \frac{1}{2}\) are polynomials. Their product will also be a polynomial. This demonstrates closure.

3. \((\frac{x^2}{2})(\frac{1}{x})\): This product results in \( \frac{x}{2} \), which is a polynomial. However, it involves division by \(x\), which doesn't show closure under multiplication of polynomials specifically.

4. \(x^2 + 2x + 1\): This is a polynomial, but it does not demonstrate a multiplication of two polynomials resulting in another polynomial.

From the analysis above, the response that demonstrates that polynomials form a closed system under multiplication is:

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

Answer 4

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

(12x2−3)(4y3+5x2)

(1 point)
Responses

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

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

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

x2−−√y3+4x4−15x2

Answer 5

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

1. Distribute \(12x^2\) across \(4y^3\) and \(5x^2\): \[ 12x^2 \cdot 4y^3 = 48x^2y^3 \] \[ 12x^2 \cdot 5x^2 = 60x^4 \]

2. Distribute \(-3\) across \(4y^3\) and \(5x^2\): \[ -3 \cdot 4y^3 = -12y^3 \] \[ -3 \cdot 5x^2 = -15x^2 \]

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

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

Now, let's check the given responses:

1. \(52x^2y^3 + 2x^4 - 8y^3 + 15x^2\)
2. \(2x^2y^3 + 52x^4 - 12y^3 - 15x^2\)
3. \(x^2y^3 + 5x^4 - 8y^3 - 15x^2\)
4. \(\sqrt{x}y^3 + 4x^4 - 15x^2\)

None of the provided responses match our product \(60x^4 + 48x^2y^3 - 15x^2 - 12y^3\).

Therefore, the calculation of the product does demonstrate closure under multiplication of polynomials, but none of the response options represent this product correctly. If any were intended to match, it appears there may have been an error in the options provided.