Which of the following shows that polynomials form a closed system under multiplication? Option #1: 3(2x+5)=6x+15 Option #2: 8x+7=5x+3x+1+6 Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24 Option #4: 2x12⋅3=6x−−√ (1 point) Option #
The correct option that shows that polynomials form a closed system under multiplication is option #3: (2x^2−4)(3y+6)=6x^2y+12x^2−12y−24.
Which of the following shows that polynomials form a closed system under multiplication?
Option #1: 3(2x+5)=6x+15
Option #2: 8x+7=5x+3x+1+6
Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24
Option #4: 2x12⋅3=6x−−√ (1 point) Option #
The correct option that shows that polynomials form a closed system under multiplication is option #3: (2x^2−4)(3y+6)=6x^2y+12x^2−12y−24.
The option that shows that polynomials form a closed system under multiplication is Option #3: (2x^2−4)(3y+6)=6x^2y+12x^2−12y−24.
In this option, we are multiplying two polynomials: (2x^2−4) and (3y+6). The result of the multiplication is expressed as another polynomial: 6x^2y+12x^2−12y−24. This demonstrates that when we multiply two polynomials, the result is still a polynomial, thus showing that polynomials form a closed system under multiplication.
To determine whether polynomials form a closed system under multiplication, we need to check if the product of any two polynomials is also a polynomial. Let's analyze each option to see if it satisfies this property:
Option #1: 3(2x+5)=6x+15
In this equation, we are multiplying a polynomial (2x+5) by a constant (3). The result is another polynomial (6x+15). Therefore, this option demonstrates that polynomials form a closed system under multiplication.
Option #2: 8x+7=5x+3x+1+6
This equation shows the addition of polynomial terms (8x, 7), not multiplication. Therefore, this option does not demonstrate the property we are looking for.
Option #3: (2x^2−4)(3y+6)=6x^2y+12x^2−12y−24
Here, we have the multiplication of two polynomials, (2x^2−4) and (3y+6). The result is also a polynomial (6x^2y+12x^2−12y−24). Thus, this option supports the idea that polynomials form a closed system under multiplication.
Option #4: 2x^12⋅3=6√x
In this option, we have a square root (√) symbol, which does not represent polynomial multiplication. Therefore, this option does not demonstrate the property we are investigating.
From the analysis, we can conclude that option #1 and option #3 show that polynomials form a closed system under multiplication.