To understand which statement accurately demonstrates how the product of [tex]\((x + 7)^2\)[/tex] shows the closure property of multiplication, let's follow a detailed, step-by-step explanation.
1. Expanding the Expression [tex]\((x + 7)^2\)[/tex]:
The expression [tex]\((x + 7)^2\)[/tex] can be expanded using the special product formula for the square of a binomial:
[tex]\[
(x + 7)^2 = (x + 7)(x + 7)
\][/tex]
By applying the distributive property (also known as the FOIL method for binomials), we get:
[tex]\[
(x + 7)(x + 7) = x(x + 7) + 7(x + 7)
= x^2 + 7x + 7x + 49
= x^2 + 14x + 49
\][/tex]
2. Resulting Polynomial:
The expansion results in the polynomial [tex]\(x^2 + 14x + 49\)[/tex].
3. Closure Property of Multiplication:
The closure property of multiplication for polynomials states that the product of any polynomials is also a polynomial.
4. Verification with Our Result:
- We started with the polynomial [tex]\(x + 7\)[/tex].
- We multiplied it by itself to get [tex]\((x + 7)^2\)[/tex], which expanded to [tex]\(x^2 + 14x + 49\)[/tex].
- The resulting expression [tex]\(x^2 + 14x + 49\)[/tex] is clearly a polynomial because it fits the definition of a polynomial (a sum of terms consisting of a variable raised to a non-negative integer power and multiplied by a coefficient).
5. Conclusion:
The correct statement that shows how the product of [tex]\((x + 7)^2\)[/tex] demonstrates the closure property of multiplication is:
[tex]\[
\boxed{x^2 + 14x + 49 \text{ is a polynomial}}
\][/tex]
