Which statement shows how the product of [tex]$(x+7)^2$[/tex] demonstrates the closure property of multiplication?

A. [tex]$x^2 + 14x + 49$[/tex] is a polynomial.
B. [tex][tex]$x^2 + 14x + 49$[/tex][/tex] may or may not be a polynomial.
C. [tex]$x^2 + 49$[/tex] is a polynomial.
D. [tex]$x^2 + 49$[/tex] may or may not be a polynomial.



Answer :

Alright, let's solve this step-by-step!

To address the question of how the product of [tex]\((x + 7)^2\)[/tex] demonstrates the closure property of multiplication among polynomials, we should start by expanding the expression [tex]\((x + 7)^2\)[/tex].

1. Expand [tex]\((x + 7)^2\)[/tex]:

When we expand [tex]\((x + 7)^2\)[/tex], we perform the multiplication as follows:
[tex]\[ (x + 7)^2 = (x + 7)(x + 7) \][/tex]
We can use the distributive property to expand:
[tex]\[ (x + 7)(x + 7) = x(x + 7) + 7(x + 7) \][/tex]
Now, distribute [tex]\(x\)[/tex] and [tex]\(7\)[/tex] across [tex]\(x + 7\)[/tex]:
[tex]\[ x(x + 7) = x^2 + 7x \][/tex]
[tex]\[ 7(x + 7) = 7x + 49 \][/tex]
Combine these results:
[tex]\[ x^2 + 7x + 7x + 49 = x^2 + 14x + 49 \][/tex]

2. Identify the Polynomial:

The result of the expansion is:
[tex]\[ x^2 + 14x + 49 \][/tex]
This expression is a polynomial because it is a sum of terms, each of which consists of a variable raised to a non-negative integer power and multiplied by a coefficient. Specifically, this is a quadratic polynomial (degree 2).

3. Closure Property:

The closure property of multiplication for polynomials states that the product of any two polynomials is also a polynomial. Since [tex]\((x + 7)^2\)[/tex] is the product of [tex]\((x + 7)\)[/tex] with itself, it must also be a polynomial. Therefore, the expression we obtained, [tex]\(x^2 + 14x + 49\)[/tex], should be a polynomial.

Given these points:
- [tex]\(x^2 + 14x + 49\)[/tex] is indeed a polynomial.

Among the given options:
- The correct statement is

[tex]\[ x^2 + 14 x + 49 \text{ is a polynomial}. \][/tex]

Thus, the correct choice is:
- [tex]\(x^2 + 14x + 49 \text{ is a polynomial}\)[/tex].

This demonstrates the closure property because the product of [tex]\((x + 7)^2\)[/tex] results in another polynomial.