{"id":"cb_question_bank:6043455d-f2c9-4035-924e-756790b2a91b","source_id":"cb_question_bank","source_item_id":"6043455d-f2c9-4035-924e-756790b2a91b","external_id":"4b8535d8-7c5d-4bfd-9256-245ea6bb505f","ibn":null,"question_id":"ffdbcad4","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The expression 4 x squared plus b x minus 45 , where b is a constant, can be rewritten as left parenthesis h x plus k right parenthesis left parenthesis x plus j right parenthesis , where h , k , and j are integer constants. Which of the following must be an integer?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The expression <math alttext=\"4 x squared plus b x minus 45\"><mrow>\n\t<mrow>\n\t\t<mn>4</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<mi>b</mi>\n\t\t<mi>x</mi>\n\t</mrow>\n\t<mo>-</mo>\n\t<mn>45</mn>\n</mrow>\n</math>, where <math alttext=\"b\"><mi>b</mi>\n</math> is a constant, can be rewritten as <math alttext=\"left parenthesis h x plus k right parenthesis left parenthesis x plus j right parenthesis\"><mfenced><mrow><mi>h</mi><mi>x</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mi>j</mi></mrow></mfenced></math>, where <math alttext=\"h\"><mi>h</mi>\n</math>, <math alttext=\"k\"><mi>k</mi>\n</math>, and <math alttext=\"j\"><mi>j</mi>\n</math> are integer constants. Which of the following must be an integer?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D is correct. It's given that <math alttext=\"4 x squared plus b x minus 45\"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>-</mo><mn>45</mn></math> can be rewritten as <math alttext=\"left parenthesis h x plus k right parenthesis left parenthesis x plus j right parenthesis\"><mfenced><mrow><mi>h</mi><mi>x</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mi>j</mi></mrow></mfenced></math>. The expression <math alttext=\"left parenthesis h x plus k right parenthesis left parenthesis x plus j right parenthesis\"><mfenced><mrow><mi>h</mi><mi>x</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mi>j</mi></mrow></mfenced></math> can be rewritten as&nbsp;<math alttext=\"h x squared plus j h x plus k x plus k j\"><mi>h</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>j</mi><mi>h</mi><mi>x</mi><mo>+</mo><mi>k</mi><mi>x</mi><mo>+</mo><mi>k</mi><mi>j</mi></math>, or&nbsp;<math alttext=\"h x squared plus left parenthesis j h plus k right parenthesis x plus k j\"><mi>h</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfenced><mrow><mi>j</mi><mi>h</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>k</mi><mi>j</mi></math>. Therefore,&nbsp;<math alttext=\"h x squared plus left parenthesis j h plus k right parenthesis x plus k j\"><mi>h</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfenced><mrow><mi>j</mi><mi>h</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>k</mi><mi>j</mi></math> is equivalent to <math alttext=\"4 x squared plus b x minus 45\"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>-</mo><mn>45</mn></math>. It follows that <math alttext=\"k j equals negative 45\"><mi>k</mi><mi>j</mi><mo>=</mo><mo>-</mo><mn>45</mn></math>. Dividing each side of this equation by <math alttext=\"k\"><mi>k</mi>\n</math> yields&nbsp;<math alttext=\"j equals StartFraction negative 45 Over k EndFraction\"><mi>j</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>45</mn></mrow><mi>k</mi></mfrac></math>. Since <math alttext=\"j\"><mi>j</mi>\n</math> is an integer, <math alttext=\"minus StartFraction 45 Over k EndFraction\"><mrow>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>45</mn>\n\t\t<mi>k</mi>\n\t</mfrac>\n</mrow>\n</math> must be an integer. Therefore, <math alttext=\"StartFraction 45 Over k EndFraction\"><mfrac>\n\t\t<mn>45</mn>\n\t\t<mi>k</mi>\n\t</mfrac>\n</math> must also be an integer.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["D"],"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><math alttext=\"StartFraction b Over h EndFraction\"><mrow>\n\t<mfrac>\n\t\t<mi>b</mi>\n\t\t<mi>h</mi>\n\t</mfrac>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"StartFraction b Over k EndFraction\"><mrow>\n\t<mfrac>\n\t\t<mi>b</mi>\n\t\t<mi>k</mi>\n\t</mfrac>\n</mrow>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"StartFraction 45 Over h EndFraction\"><mrow>\n\t<mfrac>\n\t\t<mn>45</mn>\n\t\t<mi>h</mi>\n\t</mfrac>\n</mrow>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"StartFraction 45 Over k EndFraction\"><mrow>\n\t<mfrac>\n\t\t<mn>45</mn>\n\t\t<mi>k</mi>\n\t</mfrac>\n</mrow>\n</math></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}}