To show that polynomials form a closed system under multiplication, we need an example where multiplying two polynomials results in another polynomial.
Let's analyze the options:
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Option #1: \(3(2x+5)=6x+15\)
This is an example of distributing a number (3) with a polynomial, showing one polynomial operation, but it does not illustrate multiplication of two polynomials resulting in another polynomial. -
Option #2: \(8x+7=5x+3x+1+6\)
This is simply a rearrangement and combining like terms; it does not show multiplication. -
Option #3: \((2x^2−4)(3y+6)=6x^2y+12x^2−12y−24\)
This is showing the multiplication of two polynomials \((2x^2 - 4)\) and \((3y + 6)\), which results in \((6x^2y + 12x^2 - 12y - 24)\), a polynomial. This is an example that clearly shows that the product is also a polynomial. -
Option #4: \(2x^{12} \cdot 3=6x^{12}\)
This shows multiplication of a monomial and a constant, resulting in another monomial, which is a polynomial. However, it is a simpler case.
Among all the options, Option #3 best illustrates that polynomials form a closed system under multiplication, as it involves the multiplication of two non-trivial polynomials leading to another polynomial.
Thus, the correct answer is Option #3.
3 answers