Question

if a polynomial function \\(f(x)\\) has roots \\(3+\sqrt{5}\\) and \\(-6\\), what must be a factor of \\(f(x)\\)?

\\((x+(3-\sqrt{5})\\)
\\((x-(3-\sqrt{5}))\\)
\\((x+(5+\sqrt{3}))\\)
\\((x-(5-\sqrt{3}))\\)

Explanation:

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<reasoning>

Identify the given roots and polynomial properties

Using the Roots of Polynomials knowledge point
Given roots:
\[
x_1 = 3 + \sqrt{5},\quad x_2 = -6
\]

Apply the Irrational Conjugate Roots Theorem

For a polynomial with rational coefficients, if \(3 + \sqrt{5}\) is a root, then its irrational conjugate must also be a root:
\[
x_3 = 3 - \sqrt{5}
\]

Determine the corresponding factors

Using the Factor Theorem knowledge point
\[

$$\begin{aligned} &\text{If } x = r \text{ is a root, then } (x - r) \text{ is a factor.}\\ &\text{For } x_3 = 3 - \sqrt{5} \implies (x - (3 - \sqrt{5})) \text{ must be a factor.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((x + (3 - \sqrt{5}))\)</mcq-option>
<mcq-correct>(B) \((x - (3 - \sqrt{5}))\)</mcq-correct>
<mcq-option>(C) \((x + (5 + \sqrt{3}))\)</mcq-option>
<mcq-option>(D) \((x - (5 - \sqrt{3}))\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given roots and polynomial properties

Using the Roots of Polynomials knowledge point
Given roots:
\[
x_1 = 3 + \sqrt{5},\quad x_2 = -6
\]

Apply the Irrational Conjugate Roots Theorem

For a polynomial with rational coefficients, if \(3 + \sqrt{5}\) is a root, then its irrational conjugate must also be a root:
\[
x_3 = 3 - \sqrt{5}
\]

Determine the corresponding factors

Using the Factor Theorem knowledge point
\[

$$\begin{aligned} &\text{If } x = r \text{ is a root, then } (x - r) \text{ is a factor.}\\ &\text{For } x_3 = 3 - \sqrt{5} \implies (x - (3 - \sqrt{5})) \text{ must be a factor.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((x + (3 - \sqrt{5}))\)</mcq-option>
<mcq-correct>(B) \((x - (3 - \sqrt{5}))\)</mcq-correct>
<mcq-option>(C) \((x + (5 + \sqrt{3}))\)</mcq-option>
<mcq-option>(D) \((x - (5 - \sqrt{3}))\)</mcq-option>
</answer>

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