Question

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

Explanation:

Step1: Recall Factor-Root Relationship

For a polynomial, if \( r \) is a root, then \( (x - r) \) is a factor. Also, for irrational roots like \( 3 + \sqrt{5} \), its conjugate \( 3 - \sqrt{5} \) is also a root (if the polynomial has rational coefficients, which is implied here as we're looking for factors).

Step2: Analyze Each Option

• For root \( -6 \), the factor would be \( (x - (-6))=(x + 6) \), but this isn't an option. For root \( 3+\sqrt{5} \), its conjugate \( 3 - \sqrt{5} \) is a root, so the factor is \( (x-(3 - \sqrt{5})) \). Let's check options:
• Option 1: \( (x+(3 - \sqrt{5}))=(x - (-(3 - \sqrt{5}))) \), not a factor of roots we have.
• Option 2: \( (x-(3 - \sqrt{5})) \), this is the factor for root \( 3 - \sqrt{5} \) (conjugate of \( 3+\sqrt{5} \)).
• Option 3: \( (x+(5+\sqrt{3})) \), no relation to given roots.
• Option 4: \( (x-(5 - \sqrt{3})) \), no relation to given roots.

Answer:

(x − (3−√5))