To show that polynomials form a closed system under multiplication, we need to demonstrate that the product of two polynomials is also a polynomial.
Among the options provided, let's analyze each one for polynomial multiplication:
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Option #1: \(3(2x+5)=6x+15\)
- This shows the multiplication of a constant (3) and a polynomial (2x + 5). The result is indeed a polynomial. However, this doesn't demonstrate the multiplication of two polynomials.
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Option #2: \(8x+7=5x+3x+1+6\)
- This is an equation representing a polynomial simplification, but it does not show multiplication, so it doesn't demonstrate closure under multiplication.
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Option #3: \((2x^2-4)(3y+6)=6x^2y+12x^2−12y−24\)
- This shows the multiplication of two polynomials \((2x^2 - 4)\) and \((3y + 6)\). The resulting expression is also a polynomial, which demonstrates closure under multiplication.
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Option #4: \(2x^{12} \cdot 3=6x^{12}\)
- Similar to option #1, this is the multiplication of a constant (3) and a polynomial (\(2x^{12}\)), resulting in a polynomial. However, like option #1, it does not show the multiplication of two polynomials.
Conclusion: The option that demonstrates polynomials forming a closed system under multiplication is Option #3: \((2x^2−4)(3y+6)=6x^2y+12x^2−12y−24\).
3 answers