{"id":"cb_question_bank:14c4f302-14cc-4bd0-b52a-ff9ee5ef96ec","source_id":"cb_question_bank","source_item_id":"14c4f302-14cc-4bd0-b52a-ff9ee5ef96ec","external_id":"77bede4f-84ec-4063-9b73-a0d1bd0cff6b","ibn":null,"question_id":"c3b116d7","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Which of the following expressions is(are) a factor of 3 x squared plus 20 x minus 63 ? x minus 9 3 x minus 7","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">Which of the following expressions is(are) a factor of <math alttext=\"3 x squared plus 20 x minus 63\"><mrow>\n\t<mrow>\n\t\t<mn>3</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>20</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n\t<mo>-</mo>\n\t<mn>63</mn>\n</mrow>\n</math>?</p>\n<ol style=\"list-style-type: upper-roman;\">\n<li>&nbsp;<math alttext=\"x minus 9\"><mrow>\n\t<mi>x</mi>\n\t<mo>-</mo>\n\t<mn>9</mn>\n</mrow>\n</math></li>\n<li><math alttext=\"3 x minus 7\"><mrow>\n\t<mrow>\n\t\t<mn>3</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n\t<mo>-</mo>\n\t<mn>7</mn>\n</mrow>\n</math></li>\n</ol>","rationale_html":"<p>Choice B is correct. The given expression can be factored by first finding two values whose sum is <math alttext=\"20\"><mn>20</mn>\n</math> and whose product is <math alttext=\"3 left parenthesis negative 63 right parenthesis\"><mn>3</mn><mfenced><mrow><mo>-</mo><mn>63</mn></mrow></mfenced></math>, or <math alttext=\"negative 189\"><mo>-</mo><mn>189</mn>\n</math>. Those two values are <math alttext=\"27\"><mn>27</mn>\n</math> and <math alttext=\"negative 7\"><mo>-</mo><mn>7</mn>\n</math>. It follows that the given expression can be rewritten as <math alttext=\"3 x squared plus 27 x minus 7 x minus 63\"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>27</mn><mi>x</mi><mo>-</mo><mn>7</mn><mi>x</mi><mo>-</mo><mn>63</mn></math>. Since the first two terms of this expression have a common factor of <math alttext=\"3 x\"><mrow>\n\t<mn>3</mn>\n\t<mi>x</mi>\n</mrow>\n</math> and the last two terms of this expression have a common factor of <math alttext=\"negative 7\"><mo>-</mo><mn>7</mn>\n</math>, this expression can be rewritten as <math alttext=\"3 x left parenthesis x plus 9 right parenthesis minus 7 left parenthesis x plus 9 right parenthesis\"><mn>3</mn><mi>x</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mfenced><mo>-</mo><mn>7</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mfenced></math>. Since the two terms of this expression have a common factor of <math alttext=\"left parenthesis x plus 9 right parenthesis\"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mfenced></math>, it can be rewritten as <math alttext=\"left parenthesis 3 x minus 7 right parenthesis left parenthesis x plus 9 right parenthesis\"><mfenced><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>7</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mfenced></math>. Therefore, expression II,&nbsp;<math alttext=\"3 x minus 7\"><mn>3</mn><mi>x</mi><mo>-</mo><mn>7</mn></math>, is a factor of <math alttext=\"3 x squared plus 20 x minus 63\"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>20</mn><mi>x</mi><mo>-</mo><mn>63</mn></math>, but expression I,&nbsp;<math alttext=\"x minus 9\"><mi>x</mi><mo>-</mo><mn>9</mn></math>, is not a factor of <math alttext=\"3 x squared plus 20 x minus 63\"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>20</mn><mi>x</mi><mo>-</mo><mn>63</mn></math>.&nbsp;</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["B"],"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":[{"label":"A","content_html":"<p style=\"text-align: left;\">I only</p>","ord":0},{"label":"B","content_html":"<p>II only</p>","ord":1},{"label":"C","content_html":"<p style=\"text-align: left;\">I and II</p>","ord":2},{"label":"D","content_html":"<p style=\"text-align: left;\">Neither I nor II</p>","ord":3}],"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}}