Ask a New Question
Menu

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 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

1 answer

To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two polynomials \( (12x^2 - 3) \) and \( (4y^3 + 5x^2) \) and find the resultant polynomial.

Let's calculate the multiplication step-by-step:

  1. Distribute \( 12x^2 \) through \( (4y^3 + 5x^2) \):

    • \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
    • \( 12x^2 \cdot 5x^2 = 60x^4 \)

  2. Distribute \( -3 \) through \( (4y^3 + 5x^2) \):

    • \( -3 \cdot 4y^3 = -12y^3 \)
    • \( -3 \cdot 5x^2 = -15x^2 \)

  3. Now combine all the terms: \[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]

  4. Rearranging the terms, we have: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

Thus, the product is: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

This polynomial shows that the product of two polynomials is another polynomial, demonstrating the Closure Property of multiplication of polynomials. The correct representation from your provided options may not exist here, but this is the accurate result of the multiplication.