{"id":13299,"date":"2023-05-01T10:00:18","date_gmt":"2023-05-01T04:30:18","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=13299"},"modified":"2023-05-02T10:20:05","modified_gmt":"2023-05-02T04:50:05","slug":"partial-fractions","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/partial-fractions\/","title":{"rendered":"Partial Fractions"},"content":{"rendered":"<h2>Partial Fractions<\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/PYHIAsAMqoc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128758\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-1.jpg\" alt=\"Partial Fractions 1\" width=\"932\" height=\"416\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-1.jpg 932w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-1-300x134.jpg 300w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-1-768x343.jpg 768w\" sizes=\"(max-width: 932px) 100vw, 932px\" \/><\/p>\n<p>An expression of the form \\(\\frac { f(x) }{ g(x) }\\), where f(x) and g(x) are polynomial in x, is called a rational fraction.<\/p>\n<ol>\n<li><strong>Proper rational functions:<\/strong> Functions of the form \\(\\frac { f(x) }{ g(x) }\\), where f(x) and g(x) are polynomials and g(x) \u2260 0, are called rational functions of x.<br \/>\nIf degree of f(x) is less than degree of g(x),then is called a proper rational function.<\/li>\n<li><strong>Improper rational functions:<\/strong> If degree of f(x) is greater than or equal to degree of g(x), then \\(\\frac { f(x) }{ g(x) }\\)\u00a0is called an improper rational function.<\/li>\n<li><strong>Partial fractions:<\/strong> Any proper rational function can be broken up into a group of different rational fractions, each having a simple factor of the denominator of the original rational function. Each such fraction is called a partial fraction.<\/li>\n<\/ol>\n<p>If by some process, we can break a given rational function \\(\\frac { f(x) }{ g(x) }\\)\u00a0into different fractions, whose denominators are the factors of g(x),then the process of obtaining them is called the resolution or decomposition of \\(\\frac { f(x) }{ g(x) }\\) into its partial fractions.<\/p>\n<h3>Different cases of partial fractions<\/h3>\n<p><strong>(1) When the denominator consists of non-repeated linear factors:<\/strong><br \/>\nTo each linear factor (x &#8211; a) occurring once in the denominator of a proper fraction, there corresponds a single partial fraction of the form \\(\\frac { A }{ x-a }\\), where A is a constant to be determined.<br \/>\nIf g(x) = (x \u2013 a<sub>1<\/sub>)(x \u2013 a<sub>2<\/sub>)(x \u2013 a<sub>3<\/sub>) \u2026\u2026. (x \u2013 a<sub>n<\/sub>), then we assume that,<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128759\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-2.jpg\" alt=\"Partial Fractions 2\" width=\"322\" height=\"60\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-2.jpg 322w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-2-300x56.jpg 300w\" sizes=\"(max-width: 322px) 100vw, 322px\" \/><br \/>\nwhere A<sub>1<\/sub>, A<sub>2<\/sub>, A<sub>3<\/sub>, \u2026\u2026\u2026. A<sub>n\u00a0<\/sub>are constants, can be determined by equating the numerator of L.H.S. to the numerator of R.H.S. (after L.C.M.) and substituting x = a<sub>1<\/sub>, a<sub>2<\/sub>,\u2026\u2026 a<sub>n<\/sub>.<br \/>\n<strong>(2) When the denominator consists of linear factors, some repeated:<\/strong><br \/>\nTo each linear factor (x \u2013 a) occurring r times in the denominator of a proper rational function, there corresponds a sum of r partial fractions.<br \/>\nLet g(x) = (x \u2013 a)<sup>k<\/sup>(x \u2013 a<sub>1<\/sub>)(x \u2013 a<sub>2<\/sub>) \u2026\u2026. (x \u2013 a<sub>r<\/sub>). Then we assume that<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128761\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-3.jpg\" alt=\"Partial Fractions 3\" width=\"519\" height=\"58\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-3.jpg 519w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-3-300x34.jpg 300w\" sizes=\"(max-width: 519px) 100vw, 519px\" \/><br \/>\nWhere A<sub>1<\/sub>, A<sub>2<\/sub>, A<sub>3<\/sub>, \u2026\u2026\u2026. A<sub>k\u00a0<\/sub>are constants. To determine the value of constants adopt the procedure as above.<br \/>\n<strong>(3) When the denominator consists of non-repeated quadratic factors:<\/strong><br \/>\nTo each irreducible non repeated quadratic factor ax<sup>2<\/sup> + bx + c, there corresponds a partial fraction of the form \\(\\frac { Ax+B }{ { a{ x }^{ 2 }+bx+c } }\\), where A and B are constants to be determined.<br \/>\nExample :<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128763\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-4.jpg\" alt=\"Partial Fractions 4\" width=\"462\" height=\"243\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-4.jpg 462w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-4-300x158.jpg 300w\" sizes=\"(max-width: 462px) 100vw, 462px\" \/><br \/>\n<strong>(4) When the denominator consists of repeated quadratic factors:<\/strong><br \/>\nTo each irreducible quadratic factor ax<sup>2<\/sup> + bx + c occurring r times in the denominator of a proper rational fraction there corresponds a sum of r partial fractions of the form.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128764\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-5.jpg\" alt=\"Partial Fractions 5\" width=\"468\" height=\"63\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-5.jpg 468w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Partial-Fractions-5-300x40.jpg 300w\" sizes=\"(max-width: 468px) 100vw, 468px\" \/><br \/>\nwhere, A\u2019s and B\u2019s are constants to be determined.<\/p>\n<h3>Partial fractions of improper rational functions<\/h3>\n<p>If degree of is greater than or equal to degree of g(x), then \\(\\frac { f(x) }{ g(x) }\\) is called an <strong>improper rational function<\/strong> and every rational function can be transformed to a proper rational function by dividing the numerator by the denominator.<br \/>\nWe divide the numerator by denominator until a remainder is obtained which is of lower degree than the denominator.<\/p>\n<h3>General method of finding out the constants<\/h3>\n<ol>\n<li>Express the given fraction into its partial fractions in accordance with the rules written above.<\/li>\n<li>Then multiply both sides by the denominator of the given fraction and you will get an identity which will hold for all values of x.<\/li>\n<li>Equate the coefficients of like powers of x in the resulting identity and solve the equations so obtained simultaneously to find the various constant is short method. Sometimes, we substitute particular values of the variable x in the identity obtained after clearing of fractions to find some or all the constants. For non-repeated linear factors, the values of x used as those for which the denominator of the corresponding partial fractions become zero.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Partial Fractions An expression of the form , where f(x) and g(x) are polynomial in x, is called a rational fraction. Proper rational functions: Functions of the form , where f(x) and g(x) are polynomials and g(x) \u2260 0, are called rational functions of x. If degree of f(x) is less than degree of g(x),then [&hellip;]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[4826,4828,4825,4823,4827,4824],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Partial Fractions - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/partial-fractions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Partial Fractions\" \/>\n<meta property=\"og:description\" content=\"Partial Fractions An expression of the form , where f(x) and g(x) are polynomial in x, is called a rational fraction. 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