according to the rational roots theorem, which statement about $f(x) = 25x^7 - x^6 - 5x^4 + x - 49$ is true?

any rational root of $f(x)$ is a multiple of $-49$ divided by a multiple of $25$.
any rational root of $f(x)$ is a multiple of $25$ divided by a multiple of $-49$.
any rational root of $f(x)$ is a factor of $-49$ divided by a factor of $25$.
any rational root of $f(x)$ is a factor of $25$ divided by a factor of $-49$.

Question

according to the rational roots theorem, which statement about $f(x) = 25x^7 - x^6 - 5x^4 + x - 49$ is true?

any rational root of $f(x)$ is a multiple of $-49$ divided by a multiple of $25$.
any rational root of $f(x)$ is a multiple of $25$ divided by a multiple of $-49$.
any rational root of $f(x)$ is a factor of $-49$ divided by a factor of $25$.
any rational root of $f(x)$ is a factor of $25$ divided by a factor of $-49$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Rational Root Theorem", "Potential Rational Roots" ], "new_concepts": [], "current_concepts": [ "Rational Root Theorem", "Potential Rational Roots" ] } ## Identify the coefficients Using the *Rational Root Theorem* knowledge point $$ \begin{aligned} &f(x) = 25x^7 - x^6 - 5x^4 + x - 49\ &a_n = 25\ &a_0 = -49 \end{aligned} $$ ## Apply the theorem Using the *Potential Rational Roots* knowledge point $$ \begin{aligned} &\text{Rational root } \frac{p}{q} \text{ requires:}\ &p \text{ is a factor of } a_0 = -49\ &q \text{ is a factor of } a_n = 25 \end{aligned} $$

Any rational root of $f(x)$ is a multiple of $-49$ divided by a multiple of $25$.

Any rational root of $f(x)$ is a multiple of $25$ divided by a multiple of $-49$.

Any rational root of $f(x)$ is a factor of $-49$ divided by a factor of $25$.

Any rational root of $f(x)$ is a factor of $25$ divided by a factor of $-49$.

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify the coefficients

Apply the theorem

Answer: