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:

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

$$
LATEXBLOCK0
$$

Answer:

• (A) \((x + (3 - \sqrt{5}))\)
• (B) \((x - (3 - \sqrt{5}))\) (Correct answer)
• (C) \((x + (5 + \sqrt{3}))\)
• (D) \((x - (5 - \sqrt{3}))\)