To determine which statement correctly demonstrates the closure property under multiplication for the polynomials [tex]\(3x + 6\)[/tex] and [tex]\(5x^2 - 4x\)[/tex], let's verify the result of multiplying these polynomials.
When you multiply two polynomials, the result should also be a polynomial. Here is the step-by-step process of multiplying [tex]\(3x + 6\)[/tex] and [tex]\(5x^2 - 4x\)[/tex]:
1. Distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[
(3x + 6)(5x^2 - 4x)
\][/tex]
2. Multiply the terms:
[tex]\[
3x \cdot 5x^2 = 15x^3
\][/tex]
[tex]\[
3x \cdot (-4x) = -12x^2
\][/tex]
[tex]\[
6 \cdot 5x^2 = 30x^2
\][/tex]
[tex]\[
6 \cdot (-4x) = -24x
\][/tex]
3. Combine the resulting terms:
[tex]\[
15x^3 - 12x^2 + 30x^2 - 24x
\][/tex]
4. Combine like terms:
[tex]\[
15x^3 + (30x^2 - 12x^2) - 24x = 15x^3 + 18x^2 - 24x
\][/tex]
Therefore, the product of [tex]\(3x + 6\)[/tex] and [tex]\(5x^2 - 4x\)[/tex] is [tex]\(15x^3 + 18x^2 - 24x\)[/tex], which is a polynomial. According to the closure property of polynomials, the product of two polynomials is also a polynomial.
The correct statement is:
[tex]\[
15x^3 + 18x^2 - 24x \text{ is a polynomial}
\][/tex]
