graph the system of equations to find the solutions of $x^3 + 6x^2 - 40x = 192$.

$y = x^3 + 6x^2 - 40x$

$y = 192$

select all of the solution(s) to the equation.
- $x = -8$
- $x = -4$
- $x = 0$
- $x = 6$
- $x = 192$

Question

graph the system of equations to find the solutions of $x^3 + 6x^2 - 40x = 192$.

$y = x^3 + 6x^2 - 40x$

$y = 192$

select all of the solution(s) to the equation.
- $x = -8$
- $x = -4$
- $x = 0$
- $x = 6$
- $x = 192$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Polynomial Roots", "Systems of Non-linear Equations" ], "new_concepts": [], "current_concepts": [ "Polynomial Roots", "Systems of Non-linear Equations", "Solving Equations Graphically" ] } ## Set up the polynomial equation $$ x^3 + 6x^2 - 40x - 192 = 0 $$ ## Test the given integer options using synthetic division or substitution $$ \begin{aligned} &\text{For } x = -8: \ &(-8)^3 + 6(-8)^2 - 40(-8) - 192 = -512 + 384 + 320 - 192 = 0 \ &\text{For } x = -4: \ &(-4)^3 + 6(-4)^2 - 40(-4) - 192 = -64 + 96 + 160 - 192 = 100 eq 0 \ &\text{For } x = 6: \ &(6)^3 + 6(6)^2 - 40(6) - 192 = 216 + 216 - 240 - 192 = 0 \end{aligned} $$ ## Factor the polynomial to find all real roots $$ \begin{aligned} &(x + 8)(x - 6)(x - r) = 0 \ &(x^2 + 2x - 48)(x - r) = x^3 + 6x^2 - 40x - 192 \ &\text{Comparing constant terms: } 48r = -192 \implies r = -4 \ &\text{Thus, the factored form is: } (x + 8)(x - 6)(x + 4) = 0 \ &\text{The real solutions are: } x = -8, \quad x = -4, \quad x = 6 \end{aligned} $$

(A) $x = -8$

(B) $x = -4$

(C) $x = 0$

(D) $x = 6$

(E) $x = 192$

{ "elements": [ { "type": "functiongraph", "params": [ { "js": "x*x*x + 6*x*x - 40*x", "latex": "x^3 + 6x^2 - 40x" }, -10, 8 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = x^3 + 6x^2 - 40x", "withLabel": true } }, { "type": "line", "params": [ [-10, 192], [10, 192] ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "y = 192", "withLabel": true } }, { "type": "point", "params": [ [-8, 192] ], "properties": { "name": "(-8, 192)", "size": 4, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [-4, 192] ], "properties": { "name": "(-4, 192)", "size": 4, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [6, 192] ], "properties": { "name": "(6, 192)", "size": 4, "color": "#5583F2", "withLabel": true } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Systems of Non-linear Equations" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Set up the polynomial equation

Test the given integer options using synthetic division or substitution

Factor the polynomial to find all real roots

Answer: