{"id":"cb_question_bank:b522620f-e882-4b76-8fca-da55631b072b","source_id":"cb_question_bank","source_item_id":"b522620f-e882-4b76-8fca-da55631b072b","external_id":"44b4a3f1-1823-4c9d-984b-a4ea335c95ad","ibn":null,"question_id":"42f8e4b4","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"One of the factors of 2 x cubed plus 42 x squared plus 208 x is x plus b , where b is a positive constant. What is the smallest possible value of b ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">One of the factors of <math alttext=\"2 x cubed plus 42 x squared plus 208 x\"><mrow>\n\t<mrow>\n\t\t<mn>2</mn>\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<mrow>\n\t\t<mn>42</mn>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t</mrow>\n\t<mo>+</mo>\n\t<mrow>\n\t\t<mn>208</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n</mrow>\n</math> is <math alttext=\"x plus b\"><mrow>\n\t<mi>x</mi>\n\t<mo>+</mo>\n\t<mi>b</mi>\n</mrow>\n</math>, where <math alttext=\"b\"><mi>b</mi>\n</math>&nbsp;is a positive constant. What is the smallest possible value of <math alttext=\"b\"><mi>b</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"8\"><mn>8</mn>\n</math>. Since each term of the given expression, <math alttext=\"2 x cubed plus 42 x squared plus 208 x\"><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>42</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>208</mn><mi>x</mi></math>, has a factor of <math alttext=\"2 x\"><mrow>\n\t<mn>2</mn>\n\t<mi>x</mi>\n</mrow>\n</math>, the expression can be rewritten as <math alttext=\"2 x left parenthesis x squared right parenthesis plus 2 x left parenthesis 21 x right parenthesis plus 2 x left parenthesis 104 right parenthesis\"><mn>2</mn><mi>x</mi><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced><mo>+</mo><mn>2</mn><mi>x</mi><mfenced><mrow><mn>21</mn><mi>x</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mi>x</mi><mfenced><mn>104</mn></mfenced></math>, or&nbsp;<math alttext=\"2 x left parenthesis x squared plus 21 x plus 104 right parenthesis\"><mn>2</mn><mi>x</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>21</mn><mi>x</mi><mo>+</mo><mn>104</mn></mrow></mfenced></math>. Since the values <math alttext=\"8\"><mn>8</mn>\n</math> and <math alttext=\"13\"><mn>13</mn>\n</math> have a sum of <math alttext=\"21\"><mn>21</mn>\n</math> and a product of <math alttext=\"104\"><mn>104</mn>\n</math>, the expression&nbsp;<math alttext=\"x squared plus 21 x plus 104\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>21</mn><mi>x</mi><mo>+</mo><mn>104</mn></math> can be factored as&nbsp;<math alttext=\"left parenthesis x plus 8 right parenthesis left parenthesis x plus 13 right parenthesis\"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>13</mn></mrow></mfenced></math>. Therefore, the given expression can be factored as&nbsp;<math alttext=\"2 x left parenthesis x plus 8 right parenthesis left parenthesis x plus 13 right parenthesis\"><mn>2</mn><mi>x</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>13</mn></mrow></mfenced></math>. It follows that the factors of the given expression are <math alttext=\"2\"><mn>2</mn>\n</math>, <math alttext=\"x\"><mi>x</mi>\n</math>, <math alttext=\"x plus 8\"><mrow>\n\t<mi>x</mi>\n\t<mo>+</mo>\n\t<mn>8</mn>\n</mrow>\n</math>, and <math alttext=\"x plus 13\"><mrow>\n\t<mi>x</mi>\n\t<mo>+</mo>\n\t<mn>13</mn>\n</mrow>\n</math>. Of these factors, only <math alttext=\"x plus 8\"><mrow>\n\t<mi>x</mi>\n\t<mo>+</mo>\n\t<mn>8</mn>\n</mrow>\n</math> and <math alttext=\"x plus 13\"><mrow>\n\t<mi>x</mi>\n\t<mo>+</mo>\n\t<mn>13</mn>\n</mrow>\n</math> are of the form <math alttext=\"x plus b\"><mrow>\n\t<mi>x</mi>\n\t<mo>+</mo>\n\t<mi>b</mi>\n</mrow>\n</math>, where <math alttext=\"b\"><mi>b</mi>\n</math> is a positive constant. Therefore, the possible values of <math alttext=\"b\"><mi>b</mi>\n</math> are <math alttext=\"8\"><mn>8</mn>\n</math> and <math alttext=\"13\"><mn>13</mn>\n</math>. Thus, the smallest possible value of <math alttext=\"b\"><mi>b</mi>\n</math> is <math alttext=\"8\"><mn>8</mn>\n</math>.</p>","correct_answer":["8"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.821000+00:00","source_updated_at":"2023-08-02T20:25:59.821000+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}}