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

Question

(1/2x^2−3)(4y^3+5x^2)

(1 point)
Responses

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
Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 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

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
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page 13 of 13

Answer 1

To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials:

\[ ( \frac{1}{2}x^2 - 3 )( 4y^3 + 5x^2 ) \]

We will use the distributive property (also referred to as the FOIL method in the case of binomials) to multiply the two polynomials.

1. First: \[ \frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3 \]

2. Outer: \[ \frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4 \]

3. Inner: \[ -3 \cdot 4y^3 = -12y^3 \]

4. Last: \[ -3 \cdot 5x^2 = -15x^2 \]

Now, we will combine all these results:

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

Thus, the final result is:

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

Now, comparing this with your options, the correct response is:

2x²y³ + 52x⁴ −12y³ −15x²

(Note: You might have a typographical error in presenting the second term; it should be \( \frac{5}{2}x^4 \), which reads as “52x⁴” here.)