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 of the polynomial

Using the Polynomial Roots knowledge point
Given roots:

$$ x_1 = 3 + \sqrt{5},\quad x_2 = -6 $$

Apply the Irrational Conjugate Root Theorem

Using the Conjugate Root Theorem knowledge point

$$ x_3 = 3 - \sqrt{5} $$

Determine the corresponding factors

Using the Factor Theorem knowledge point

$$
LATEXBLOCK0
$$

Match with the given options

Using the Factor Theorem knowledge point

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

Answer:

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