Sure, let's solve this step-by-step to determine which statement correctly shows how the product of \((x+5)^2\) demonstrates the closure property of multiplication.
1. First, expand the expression \((x+5)^2\):
[tex]\[
(x + 5)^2 = (x + 5)(x + 5)
\][/tex]
2. Next, use the distributive property to expand the product:
[tex]\[
(x + 5)(x + 5) = x(x + 5) + 5(x + 5)
\][/tex]
3. Distribute the terms inside the parentheses:
[tex]\[
x(x + 5) = x^2 + 5x
\][/tex]
[tex]\[
5(x + 5) = 5x + 25
\][/tex]
4. Combine all the terms together:
[tex]\[
x^2 + 5x + 5x + 25
\][/tex]
5. Combine like terms:
[tex]\[
x^2 + 10x + 25
\][/tex]
The final expanded polynomial is:
[tex]\[
x^2 + 10x + 25
\][/tex]
This shows that \((x+5)^2\) expands to \(x^2 + 10x + 25\), which is indeed a polynomial. This demonstrates the closure property of multiplication, which states that the product of two polynomials is also a polynomial.
By examining the options, we're choosing the one that correctly reflects this polynomial:
\(x^2 + 10x + 25\) is a polynomial.
Therefore, the correct statement is:
[tex]\[
x^2 + 10x + 25 \text{ is a polynomial}
\][/tex]
