if a polynomial function $f(x)$ has roots $-9$ and $7 - i$, what must be a factor of $f(x)$?

$(x - (7 + i))$
$(x - (-7 - i))$
$(x + (7 + i))$
$(x + (7 - i))$

Question

if a polynomial function $f(x)$ has roots $-9$ and $7 - i$, what must be a factor of $f(x)$?

$(x - (7 + i))$
$(x - (-7 - i))$
$(x + (7 + i))$
$(x + (7 - i))$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Complex Conjugate Root Theorem", "Factor Theorem" ], "new_concepts": [], "current_concepts": [ "Complex Conjugate Root Theorem", "Factor Theorem", "Roots of Polynomials" ] } ## Identify conjugate complex roots $$ \begin{aligned} &\text{Given root: } x_1 = 7 - i\ &\text{By Complex Conjugate Root Theorem, conjugate root: } x_2 = 7 + i \end{aligned} $$ ## Apply the Factor Theorem $$ \begin{aligned} &\text{If } r \text{ is a root of } f(x)\text{, then } (x - r) \text{ is a factor.}\ &\text{For root } x_2 = 7 + i\text{, the corresponding factor is: } (x - (7 + i)) \end{aligned} $$

(A) $(x - (7 + i))$

(B) $(x - (-7 - i))$

(C) $(x + (7 + i))$

(D) $(x + (7 - i))$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Complex Conjugate Root Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify conjugate complex roots

Apply the Factor Theorem

Answer: