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.
