Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses
x2+2x+1
x squared plus 2 x plus 1
(x−−√)(x+1)
left parenthesis Start Root x End Root right parenthesis left parenthesis x plus 1 right parenthesis
(x22)(1x)
left parenthesis Start Fraction x squared over 2 End Fraction right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis
(x2+1)(x−12)
left parenthesis x squared plus 1 right parenthesis left parenthesis x minus 1 over 2 right parenthesis
Use multiplication to demonstrate these polynomials form a closed system.
(12x2−3)(4y3+5x2)
(1 point)
Responses
52x2y3+2x4−8y3+15x2
Start Fraction 5 over 2 End Fraction x squared y cubed plus 2 x superscript 4 baseline minus 8 y cubed plus 15 x squared
x2−−√y3+4x4−15x2
Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 15 x squared
2x2y3+52x4−12y3−15x2
2 x squared y cubed plus Start Fraction 5 over 2 End Fraction x superscript 4 baseline minus 12 y cubed minus 15 x squared
x2y3+5x4−8y3−15x2
x squared y cubed plus 5 x superscript 4 baseline minus 8 y cubed minus 15 x squared
The correct response that shows that polynomials form a closed system under multiplication is:
(x2+1)(x−12)
To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to understand what it means for a system to be closed under multiplication.
A system is said to be closed under multiplication if, when you multiply any two elements from that system, the result is also an element of the same system.
In this case, we are dealing with polynomials. So, for polynomials to form a closed system under multiplication, it means that when you multiply any two polynomials, the result should also be a polynomial.
Let's analyze each of the given responses:
1. x^2+2x+1: This is a polynomial, and when you multiply it with any other polynomial, the result will be a polynomial. So, this response shows that polynomials form a closed system under multiplication.
2. (x−√)(x+1): This is not a polynomial because it contains a radical (√). Since we are specifically looking at the multiplication of polynomials, this response does not demonstrate that polynomials form a closed system under multiplication.
3. (x^2/2)(1/x): This response involves dividing one polynomial by x, which is equivalent to multiplying by 1/x. While this result is still a polynomial, the given response might be confusing due to the notation used. However, this response does show that polynomials form a closed system under multiplication.
4. (x^2+1)(x−1/2): Similar to the previous explanation, this response demonstrates the multiplication of polynomials and shows that polynomials form a closed system under multiplication.
Therefore, the correct responses that show polynomials form a closed system under multiplication are:
- x^2+2x+1
- (x^2/2)(1/x)
- (x^2+1)(x−1/2)