Which of the following responses shows that polynomials form a closed system under multiplication?
Responses
4x3⋅5
4 times x cubed times 5 - no response given
6⋅−3
6⋅−3
- no response given
3x24−1
the fraction with numerator 3 x squared and denominator 4 minus 1 - no response given
(x)(12x)
The response "4x3⋅5" shows that polynomials form a closed system under multiplication.
To determine which of the following responses shows that polynomials form a closed system under multiplication, let's analyze each response:
1. 4x^3 * 5
This is a multiplication of a polynomial (4x^3) by a constant (5), which follows the property of closure.
2. 4 times x cubed times 5 - no response given
Since no response is given, we cannot determine if this response shows closure.
3. 6 * -3
This is a multiplication of two constants, which follows the property of closure.
4. 6 * -3 - no response given
Since no response is given, we cannot determine if this response shows closure.
5. 3x^2 / 4 - 1
This is a division of a polynomial (3x^2/4) by a constant (1), not a multiplication. Therefore, it does not show closure under multiplication.
6. (x) * (12x)
This is a multiplication of two polynomials (x and 12x), which follows the property of closure.
Therefore, the responses that show that polynomials form a closed system under multiplication are:
1. 4x^3 * 5
3. 6 * -3
6. (x) * (12x)