To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials, \( (12x^2 - 3)(4y^3 + 5x^2) \).
We can use the distributive property (also known as the FOIL method for binomials) to carry out the multiplication.
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Multiply each term in the first polynomial by each term in the second polynomial:
- \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
- \( 12x^2 \cdot 5x^2 = 60x^4 \)
- \( -3 \cdot 4y^3 = -12y^3 \)
- \( -3 \cdot 5x^2 = -15x^2 \)
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Combine all the results:
Putting it all together, we have: \[ (12x^2 - 3)(4y^3 + 5x^2) = 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
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Rearranging the terms (to write it in standard form): \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
The final result is: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
Closure Property
The Closure Property states that when you perform an operation (like multiplication) on two elements from a set (in this case, polynomials), the result is also an element of the same set.
Since the result \( 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \) is a polynomial, we have demonstrated that the product of two polynomials is also a polynomial, thus confirming the Closure Property of multiplication of polynomials.
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