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Which statement shows how the two polynomials [tex]4x+6[/tex] and [tex]2x^2-8x[/tex] demonstrate the closure property when multiplied?

A. [tex]8x^3-20x^2-48x[/tex] is a polynomial
B. [tex]8x^3-20x^2+48x[/tex] may or may not be a polynomial
C. [tex]8x^2-32x^2-14x[/tex] is a polynomial
D. [tex]8x^3-32x^2+14x[/tex] may or may not be a polynomial



Answer :

GinnyAnswer
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]

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