To demonstrate the closure property of polynomials, we need to verify that the product of two polynomials is also a polynomial.
Given the two polynomials:
[tex]\[ P(x) = 4x + 6 \][/tex]
[tex]\[ Q(x) = 2x^2 - 8x \][/tex]
We want to find the product \( P(x) \cdot Q(x) \).
First, we perform the multiplication:
[tex]\[ (4x + 6)(2x^2 - 8x) \][/tex]
We distribute each term in \( 4x + 6 \) to each term in \( 2x^2 - 8x \):
[tex]\[ 4x \cdot 2x^2 + 4x \cdot (-8x) + 6 \cdot 2x^2 + 6 \cdot (-8x) \][/tex]
Calculate each term:
[tex]\[ 4x \cdot 2x^2 = 8x^3 \][/tex]
[tex]\[ 4x \cdot (-8x) = -32x^2 \][/tex]
[tex]\[ 6 \cdot 2x^2 = 12x^2 \][/tex]
[tex]\[ 6 \cdot (-8x) = -48x \][/tex]
Next, we combine the like terms:
[tex]\[ 8x^3 + (-32x^2 + 12x^2) - 48x \][/tex]
[tex]\[ 8x^3 + (-20x^2) - 48x \][/tex]
So the product is:
[tex]\[ 8x^3 - 20x^2 - 48x \][/tex]
We see that the result, \( 8x^3 - 20x^2 - 48x \), is a polynomial. Therefore, it demonstrates the closure property of polynomials, which states that the product of two polynomials is also a polynomial.
Hence, the correct statement is:
[tex]\[
8x^3 - 20x^2 - 48x \text{ is a polynomial}
\][/tex]
