To demonstrate the Closure Property of multiplication of polynomials, we will multiply the two given polynomials:
\[ (1/2 x^2 - 3)(4y^3 + 5x^2) \]
We will use the distributive property (also known as the FOIL method for binomials) to perform this multiplication:
- Multiply each term in the first polynomial by each term in the second polynomial.
\[ (1/2 x^2)(4y^3) + (1/2 x^2)(5x^2) - (3)(4y^3) - (3)(5x^2) \]
- Now calculate each of these products:
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First term: \[ (1/2 x^2)(4y^3) = 2x^2y^3 \]
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Second term: \[ (1/2 x^2)(5x^2) = \frac{5}{2} x^4 \]
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Third term: \[ (-3)(4y^3) = -12y^3 \]
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Fourth term: \[ (-3)(5x^2) = -15x^2 \]
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Now combine all the computed terms: \[ 2x^2y^3 + \frac{5}{2} x^4 - 12y^3 - 15x^2 \]
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Rewrite the expression in standard polynomial form (i.e., arrange by decreasing degree): \[ \frac{5}{2} x^4 + 2x^2y^3 - 15x^2 - 12y^3 \]
From this multiplication, we can see that the product itself is a polynomial, which demonstrates the Closure Property of multiplication of polynomials. The polynomial remains within the set of polynomials after multiplication.
Thus, the final result of multiplying the two polynomials is: \[ \frac{5}{2} x^4 + 2x^2y^3 - 15x^2 - 12y^3 \]
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