if $ (x - 5) $ is a factor of $ f(x) $, which of the following must be true?

- a root of $ f(x) $ is $ x = -5 $.
- a root of $ f(x) $ is $ x = 5 $.
- both $ x = -5 $ and $ x = 5 $ are roots of $ f(x) $.
- neither $ x = -5 $ nor $ x = 5 $ is a root of $ f(x) $.

Question

if $ (x - 5) $ is a factor of $ f(x) $, which of the following must be true?

- a root of $ f(x) $ is $ x = -5 $.
- a root of $ f(x) $ is $ x = 5 $.
- both $ x = -5 $ and $ x = 5 $ are roots of $ f(x) $.
- neither $ x = -5 $ nor $ x = 5 $ is a root of $ f(x) $.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Roots" ], "new_concepts": [ "Factor Theorem" ], "current_concepts": [ "Polynomial Roots", "Factor Theorem" ] } ## Apply the Factor Theorem $$ (x - c) \text{ is a factor of } f(x) \iff f(c) = 0 $$ ## Identify the root $$ \begin{aligned} &c = 5 \ &f(5) = 0 \implies x = 5 \text{ is a root of } f(x) \end{aligned} $$

A root of $f(x)$ is $x = -5$.

A root of $f(x)$ is $x = 5$.

Both $x = -5$ and $x = 5$ are roots of $f(x)$.

Neither $x = -5$ nor $x = 5$ is a root of $f(x)$.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Factor Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Apply the Factor Theorem

Identify the root

Answer: