The question is asking which of the responses demonstrates that the set of polynomials is closed under the operation of multiplication, meaning that when you multiply two polynomials together, the result is also a polynomial.
Let's check each response:
1. 4x^3 ⋅ 5 (or 4 x cubed times 5)
This is an example of a polynomial of degree 3 being multiplied by a constant (which is technically a polynomial of degree 0). The result is \( 20x^3 \), which is still a polynomial.
2. 6 ⋅ (−3) (or 6 times negative 3)
This is an example of two constants being multiplied. While constants are polynomials of degree 0, this example doesn't really help demonstrate that polynomials are closed under multiplication, as it doesn't involve any variables (which would be present in non-constant polynomials).
3. (x) \( \left( \frac{1}{2}x \right) \) (or left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis)
This is the multiplication of two monomials (polynomials with only one term), which results in \( \frac{1}{2}x^2 \). Here we see that when multiplying these polynomials, we still get a polynomial.
4. \( \frac{3x^2}{4} − 1 \) (or Start Fraction 3 x squared over 4 minus 1 End Fraction)
This example is of a rational expression, not a multiplication of polynomials. While it may represent a polynomial (if it simplifies to one), it's not a clear example of polynomials being multiplied together.
The response that best shows polynomials form a closed system under multiplication is the third one:
(x)(\( \frac{1}{2}x \)) because it clearly demonstrates that multiplying two polynomial terms results in another polynomial.
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses
4x3⋅5
4 x cubed times 5
6⋅−3
6 times negative 3
(x)(12x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis
3x24−1
Start Fraction 3 x squared over 4 minus 1 End Fraction
1 answer