use synthetic division to solve $(x^3 - x^2 - 17x - 15) \div (x - 5)$. what is the quotient?

$x^2 + 4x + 3$
$x^2 - 6x + 13 - \frac{80}{x - 5}$
$x^3 + 4x^2 + 3x$
$x^2 - 6x + 13 - \frac{80}{x + 5}$

Question

use synthetic division to solve $(x^3 - x^2 - 17x - 15) \div (x - 5)$. what is the quotient?

$x^2 + 4x + 3$
$x^2 - 6x + 13 - \frac{80}{x - 5}$
$x^3 + 4x^2 + 3x$
$x^2 - 6x + 13 - \frac{80}{x + 5}$

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": [ "Synthetic Division" ], "current_concepts": [ "Polynomial Division", "Synthetic Division" ] } ## Set up synthetic division Using the *Polynomial Division* knowledge point $$ \begin{aligned} &\text{Dividend: } x^3 - x^2 - 17x - 15 \implies \text{Coefficients: } [1, -1, -17, -15]\ &\text{Divisor: } x - 5 \implies c = 5 \end{aligned} $$ ## Perform synthetic division calculations We set up the synthetic division table with $c = 5$ and the coefficients: $$ \begin{array}{r|rrrr} 5 & 1 & -1 & -17 & -15 \ & & 5 & 20 & 15 \ \hline & 1 & 4 & 3 & 0 \end{array} $$ ## Interpret the quotient and remainder The bottom row coefficients $[1, 4, 3, 0]$ represent: $$ \begin{aligned} &\text{Quotient: } 1x^2 + 4x + 3 = x^2 + 4x + 3\ &\text{Remainder: } 0 \end{aligned} $$

(A) $x^2+4x+3$

(B) $x^2-6x+13-\frac{80}{x-5}$

(C) $x^3+4x^2+3x$

(D) $x^2-6x+13-\frac{80}{x+5}$

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

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QUESTION IMAGE

Question

Explanation:

Set up synthetic division

Perform synthetic division calculations

Interpret the quotient and remainder

Answer: