what is the remainder when $(x^4 + 36)$ is divided by $(x^2 - 8)$?

36
64
100
117

Question

what is the remainder when $(x^4 + 36)$ is divided by $(x^2 - 8)$?

36
64
100
117

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 Division" ], "new_concepts": [], "current_concepts": [ "Polynomial Division", "Polynomial Remainder" ] } ## Set up the division problem Using the **Polynomial Division** knowledge point $$ \frac{x^4 + 36}{x^2 - 8} $$ ## Perform algebraic substitution Using the **Polynomial Division** knowledge point $$ \begin{aligned} &u = x^2\ &\frac{u^2 + 36}{u - 8} \end{aligned} $$ ## Divide the polynomials Using the **Polynomial Division** knowledge point $$ \begin{aligned} \frac{u^2 + 36}{u - 8} &= \frac{(u^2 - 64) + 100}{u - 8}\ &= \frac{(u - 8)(u + 8) + 100}{u - 8}\ &= u + 8 + \frac{100}{u - 8} \end{aligned} $$ ## Substitute back to find the remainder Using the **Polynomial Division** knowledge point $$ \begin{aligned} x^4 + 36 &= (x^2 + 8)(x^2 - 8) + 100\ \text{Remainder} &= 100 \end{aligned} $$

36

64

100

117

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

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QUESTION IMAGE

Question

Explanation:

Set up the division problem

Perform algebraic substitution

Divide the polynomials

Substitute back to find the remainder

Answer: