To determine which of the options shows that polynomials form a closed system under multiplication, we need to see if the product of the expressions given in each option results in a polynomial.
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Option #1: \( 3(2/x+5) = 6/x + 15 \)
This expression involves a term \( 2/x \), which is not a polynomial since it contains a variable in the denominator. Therefore, Option #1 does not show a closed system. -
Option #2: \( 8x + 7 = 5x + 3x + 1 + 6 \)
This equation shows that both sides are simplified polynomial expressions (the right side simplifies to \( 8x + 7 \)). However, this does not involve multiplication of polynomials. -
Option #3: \( (2x^2−4)(3y+6) = 6x^2y + 12x^2 - 12y - 24 \)
Here, two polynomials, \( 2x^2 - 4 \) and \( 3y + 6 \), are multiplied, and the product is correctly presented as another polynomial. This shows that the multiplication of two polynomials results in another polynomial. -
Option #4: \( 2x^{1/2} * 3 = 6\sqrt{x} \)
This expression involves \( x^{1/2} \), which represents a square root and is not a polynomial (as polynomials only have non-negative integer exponents). Therefore, Option #4 does not show closure under multiplication.
Given this analysis, the correct option that demonstrates that polynomials form a closed system under multiplication is:
Option #3
3 answers