-level only) Flashcards","topicId":"577863ea-188a-4cc1-9fba-a7c9ed1ed82c","topicSlug":"partial-fractions","cards":[{"id":"06f80e9f-8430-41c7-93a0-30cb6335410f","cardType":"basic","front":"How can you tell whether a rational function is proper?","back":"The numerator's degree is less than the denominator's degree.","points":1,"mcqOptions":null},{"id":"b7a49116-ec2a-47c6-a762-255337489e77","cardType":"basic","front":"What must you do before decomposing an improper rational function?","back":"Use polynomial division.","points":1,"mcqOptions":null},{"id":"a7126386-4d93-442a-bfb5-dcb0f7a02f8c","cardType":"cloze","front":"A rational function is improper when the numerator's degree is [ ] the denominator's degree.","back":"A rational function is improper when the numerator's degree is greater than or equal to the denominator's degree.","points":null,"mcqOptions":null},{"id":"a095f092-2297-448a-87b5-f4415a3cafd6","cardType":"basic","front":"What determines the required form of a partial-fraction decomposition?","back":"The factors of the denominator.","points":1,"mcqOptions":null},{"id":"e1acb78c-444d-46ee-af8a-c13b6728d374","cardType":"basic","front":"What numerator is used above a linear factor?","back":"A constant.","points":1,"mcqOptions":null},{"id":"33346321-436a-4ab8-82f9-2951a323054f","cardType":"cloze","front":"For two distinct linear factors, use $\\text{[...]}$.","back":"For two distinct linear factors, use $\\boxed{\\frac{A}{ax+b}+\\frac{B}{cx+d}}$.","points":null,"mcqOptions":null},{"id":"fec86c15-d836-4c20-8ad5-c589b69c467b","cardType":"basic","front":"What partial-fraction form is required for three distinct factors $(x-a)(x-b)(x-c)$?","back":"$$$\\frac{A}{x-a}+\\frac{B}{x-b}+\\frac{C}{x-c}$$","points":1,"mcqOptions":null},{"id":"098e3ae0-64b0-4505-a29a-6adfb27e28c7","cardType":"basic","front":"After clearing denominators, which substitutions usually isolate coefficients for linear factors?","back":"The roots of the linear factors.","points":1,"mcqOptions":null},{"id":"307fa6d0-f172-417a-9d84-63d7ea12ab70","cardType":"basic","front":"What terms are required for a repeated factor $(x-a)^2$?","back":"$$$\\frac{A}{x-a}+\\frac{B}{(x-a)^2}$$","points":1,"mcqOptions":null},{"id":"2e523829-f385-43cf-94f1-d45a307dae7f","cardType":"cloze","front":"For a repeated linear factor, include a fraction for [ ].","back":"For a repeated linear factor, include a fraction for every power of the factor.","points":null,"mcqOptions":null},{"id":"9a3cf69d-92ae-43de-bd09-3e9eb0eee29d","cardType":"basic","front":"Why is $\\frac{A}{(x-a)^2}$ alone incomplete for a squared linear factor?","back":"The term over $x-a$ is also required.","points":2,"mcqOptions":null},{"id":"51e2ed66-baf3-4ab0-b4b2-b95fa2a599c7","cardType":"basic","front":"What numerator is required above an irreducible quadratic factor?","back":"A linear numerator, $Ax+B$.","points":1,"mcqOptions":null},{"id":"5a76c68c-c730-48d9-9364-fa820fb90613","cardType":"basic","front":"Why must the numerator above a quadratic factor be linear?","back":"It must be general but lower-degree than the denominator.","points":2,"mcqOptions":null},{"id":"ad7daecb-fd8f-4e8b-a946-aff385716421","cardType":"cloze","front":"For denominator $(x-a)(x^2+b)$, use $\\text{[...]}$.","back":"For denominator $(x-a)(x^2+b)$, use $\\boxed{\\frac{A}{x-a}+\\frac{Bx+C}{x^2+b}}$.","points":null,"mcqOptions":null},{"id":"69afeb6c-a95a-490d-8283-59bbc49ba67f","cardType":"basic","front":"After clearing denominators, how can coefficients be found without convenient roots?","back":"Expand and compare coefficients of matching powers of $x$.","points":2,"mcqOptions":null},{"id":"0a5995d6-a829-4fc6-b5f0-5acb9f0ed9f7","cardType":"cloze","front":"An equation

1.2.25 Partial fractions (A-level only) Flashcards

1.2.25 Partial fractions (A-level only) Flashcards