Which of the following responses shows that polynomials form a closed system under multiplication?
13)(x4−2)
left parenthesis Start Fraction 1 over 3 End Fraction right parenthesis left parenthesis Start Fraction x over 4 End Fraction minus 2 right parenthesis
5x⋅2
5 x times 2
2x3+x2
2 x cubed plus x squared
5⋅3
5 times 3
Remaining Attempts : 3
The correct response that shows that polynomials form a closed system under multiplication is:
2x^3 + x^2
It was a
I apologize for the incorrect response.
The correct response that shows that polynomials form a closed system under multiplication is:
5x⋅2
This equation represents multiplying a polynomial (5x) by a constant (2), which still results in a polynomial.
What must be true about this multiplication problem?
(9x3+7/6)(x−2/5)
The result is not a polynomial.
The result is not a polynomial.
The result is unknown.
The result is unknown.
The result is not a closed system.
The result is not a closed system.
The result is a polynomial.
The correct answer is:
The result is a polynomial.
When multiplying two polynomials, the result will always be a polynomial.
The correct response that shows polynomials form a closed system under multiplication is 2x^3 + x^2.
In order to determine whether polynomials form a closed system under multiplication, we need to check if the product of two polynomials is still a polynomial.
Let's go through the given responses one by one and determine if they represent polynomials:
1) (x^4 - 2):
This expression represents a polynomial. It has a power of 4 on x and a constant term (-2).
2) (1/3)(x/4 - 2):
This expression also represents a polynomial. The constant term (-2) doesn't affect whether we have a polynomial or not. The fraction coefficients (1/3 and 1/4) are still considered constants for a polynomial.
3) 5x⋅2:
This expression represents a polynomial. It has a power of 1 on x (x^1) and a constant coefficient (5 * 2 = 10).
4) 2x^3 + x^2:
This expression represents a polynomial. It has a power of 3 on x (x^3), a power of 2 on x (x^2), and constant coefficients (2 and 1).
5) 5⋅3:
This expression is not a polynomial. It is a simple multiplication between two constants, which is not considered a polynomial.
Therefore, out of the given responses, the ones that show that polynomials form a closed system under multiplication are:
1) (x^4 - 2)
2) (1/3)(x/4 - 2)
3) 5x⋅2
4) 2x^3 + x^2
Response 5) 5⋅3 does not show that polynomials form a closed system under multiplication.