{"id":"cb_question_bank:e543fbaa-e0be-4f63-abfe-f1decb3f5fdc","source_id":"cb_question_bank","source_item_id":"e543fbaa-e0be-4f63-abfe-f1decb3f5fdc","external_id":"afeb4dfe-3156-4234-af28-242e9b1500f8","ibn":null,"question_id":"1dd13816","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"M","score_band":5,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"left parenthesis 5 x cubed minus 3 right parenthesis minus left parenthesis minus 4 x cubed plus 8 right parenthesis The given expression is equivalent to b x cubed minus 11 , where b is a constant. What is the value of b ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"left parenthesis 5 x cubed minus 3 right parenthesis minus left parenthesis minus 4 x cubed plus 8 right parenthesis\"><mo>(</mo><mrow><mn>5</mn></mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mrow><mn>3</mn></mrow><mo>)</mo><mo>-</mo><mo>(</mo><mo>-</mo><mrow><mn>4</mn></mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mrow><mn>8</mn></mrow><mo>)</mo></math></p>\n<p style=\"text-align: left;\">The given expression is equivalent to <math alttext=\"b x cubed minus 11\"><mrow>\n\t<mrow>\n\t\t<mi>b</mi>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>3</mn>\n\t\t</msup>\n\t</mrow>\n\t<mo>-</mo>\n\t<mn>11</mn>\n</mrow>\n</math>, where <math alttext=\"b\"><mi>b</mi>\n</math> is a constant. What is the value of <math alttext=\"b\"><mi>b</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"9\"><mn>9</mn>\n</math>. The given expression can be rewritten as <math alttext=\"left parenthesis 5 x cubed minus 3 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis minus 4 x cubed plus 8 right parenthesis\"><mfenced><mrow><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>8</mn></mrow></mfenced></math>. By applying the distributive property, this expression can be rewritten as <math alttext=\"5 x cubed minus 3 plus 4 x cubed plus left parenthesis negative 8 right parenthesis\"><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>+</mo><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfenced><mrow><mo>-</mo><mn>8</mn></mrow></mfenced></math>, which is equivalent to&nbsp;<math alttext=\"left parenthesis 5 x cubed plus 4 x cubed right parenthesis plus left parenthesis negative 3 plus left parenthesis negative 8 right parenthesis right parenthesis\"><mfenced><mrow><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>3</mn><mo>+</mo><mfenced><mrow><mo>-</mo><mn>8</mn></mrow></mfenced></mrow></mfenced></math>. Adding like terms in this expression yields&nbsp;<math alttext=\"9 x cubed minus 11\"><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>11</mn></math>. Since it's given that&nbsp;<math alttext=\"left parenthesis 5 x cubed minus 3 right parenthesis minus left parenthesis minus 4 x cubed plus 8 right parenthesis\"><mfenced><mrow><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>8</mn></mrow></mfenced></math> is equivalent to <math alttext=\"b x cubed minus 11\"><mi>b</mi><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>11</mn></math>, it follows that <math alttext=\"9 x cubed minus 11\"><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>11</mn></math> is equivalent to <math alttext=\"b x cubed minus 11\"><mi>b</mi><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>11</mn></math>. Therefore, the coefficients of <math alttext=\"x cubed\"><msup><mi>x</mi><mn>3</mn></msup></math> in these two expressions must be equivalent, and the value of <math alttext=\"b\"><mi>b</mi>\n</math> must be <math alttext=\"9\"><mn>9</mn>\n</math>.</p>","correct_answer":["9"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.822000+00:00","source_updated_at":"2023-08-02T20:25:59.822000+00:00","options":[],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}