what is the remainder when $(x^3 + 1)$ is divided by $(x^2 - x + 1)$?

$x + 1$
$x$
2
0

Question

what is the remainder when $(x^3 + 1)$ is divided by $(x^2 - x + 1)$?

$x + 1$
$x$
2
0

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": [ "Polynomial Factoring", "Polynomial Division" ], "new_concepts": [], "current_concepts": [ "Polynomial Factoring", "Polynomial Division", "Remainder Theorem" ] } ## Factor the numerator polynomial Using the *Polynomial Factoring* knowledge point $$ x^3 + 1 = (x + 1)(x^2 - x + 1) $$ ## Perform the division to find the remainder Using the *Polynomial Division* knowledge point $$ \frac{x^3 + 1}{x^2 - x + 1} = \frac{(x + 1)(x^2 - x + 1)}{x^2 - x + 1} = x + 1 $$ Since $x^2 - x + 1$ divides $x^3 + 1$ perfectly with a quotient of $x + 1$, the remainder is $0$.

(A) $x + 1$

(B) $x$

(C) $2$

(D) $0$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Division" ] }

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QUESTION IMAGE

Question

Explanation:

Factor the numerator polynomial

Perform the division to find the remainder

Answer: