To demonstrate that polynomials form a closed system under multiplication, we need to show that when we multiply any two polynomials, the result is still a polynomial.
Let's analyze the provided responses:
-
\(14x^3 \cdot 5\): This is a multiplication of a polynomial \(14x^3\) (which is a polynomial of degree 3) by a constant \(5\) (also a polynomial of degree 0). The result \(70x^3\) is a polynomial, so this shows closure under multiplication.
-
\(\frac{3x^2}{4} - 1\): This is not a multiplication but rather a polynomial expression. It does not demonstrate the closure property.
-
\((\sqrt{x})(6x \cdot -3)\): Although \(6x \cdot -3\) is polynomial (specifically, \(-18x\)), \(\sqrt{x}\) is not a polynomial, as polynomials cannot have fractional exponents. Hence, this does not demonstrate closure.
-
\((x)(12x)\): This is the multiplication of \(x\) (a polynomial of degree 1) by \(12x\) (also a polynomial of degree 1). The result is \(12x^2\), which is also a polynomial. This shows closure under multiplication.
The best responses showing that polynomials form a closed system under multiplication are:
- \(14x^3 \cdot 5\)
- \((x)(12x)\)
Both of these responses demonstrate closure since the result is still a polynomial after multiplication.
1 answer