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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]\(x^2 + 14x + 49\)[/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.


Sagot :

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]
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