We are asked to find a polynomial that has [tex]\(5\)[/tex] as its root. Let's follow the steps to arrive at the correct polynomial equation clearly.
1. Identify the Root and Form the Factor:
- Given that the solution set contains the number [tex]\(5\)[/tex], this means [tex]\(5\)[/tex] is a root of the polynomial.
- Therefore, [tex]\((x - 5)\)[/tex] is a factor of the polynomial.
2. Form a Quadratic Polynomial:
- For simplicity, we can consider the case where the polynomial is of degree 2 (a quadratic polynomial).
- The polynomial can be expressed as the square of the linear factor: [tex]\((x - 5)^2\)[/tex].
3. Expand the Polynomial:
- To expand [tex]\((x - 5)^2\)[/tex], use the binomial expansion formula: [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
- Substituting [tex]\(a = x\)[/tex] and [tex]\(b = 5\)[/tex]:
[tex]\[
(x - 5)^2 = x^2 - 2 \cdot x \cdot 5 + 5^2
\][/tex]
- Simplify the expression:
[tex]\[
(x - 5)^2 = x^2 - 10x + 25
\][/tex]
4. Verify the Form:
- We have now found that the expanded form of [tex]\((x - 5)^2\)[/tex] results in the quadratic polynomial:
[tex]\[
x^2 - 10x + 25
\][/tex]
5. Select the Correct Answer:
- Now match the expanded form with the available options:
[tex]\[
\begin{array}{l}
x^2 - 5x + 25 = 0 \\
x^2 + 10x + 25 = 0 \\
x^2 - 10x + 25 = 0 \\
x^2 + 5x + 25 = 0
\end{array}
\][/tex]
- The polynomial that matches [tex]\(x^2 - 10x + 25\)[/tex] is the correct one.
Thus, the correct polynomial equation for which [tex]\(\{5\}\)[/tex] is the solution set is:
[tex]\[
x^{\wedge} 2-10 x+25 = 0
\][/tex]
