{"id":155468,"date":"2022-03-26T00:40:48","date_gmt":"2022-03-25T19:10:48","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/partial-fractions-types-formulas-examples-and-solutions\/"},"modified":"2023-01-02T12:53:11","modified_gmt":"2023-01-02T07:23:11","slug":"partial-fractions-types-formulas-examples-and-solutions","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/","title":{"rendered":"Partial Fractions &#8211; Types, Formulas, Examples and Solutions"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/#What_are_Partial_Fractions\" title=\"What are Partial Fractions?\">What are Partial Fractions?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/#Partial_Fractions_from_Rational_Functions\" title=\"Partial Fractions from Rational Functions\">Partial Fractions from Rational Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/#Proper_Fraction_of_Improper_Function\" title=\"Proper Fraction of Improper Function\">Proper Fraction of Improper Function<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/#Heres_How_to_Solve_Partial_Fractions\" title=\"Here\u2019s How to Solve Partial Fractions!\">Here\u2019s How to Solve Partial Fractions!<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/#Now_What_are_Proper_Rational_Expressions\" title=\"Now, What are Proper Rational Expressions?\">Now, What are Proper Rational Expressions?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/#What_is_the_Degree\" title=\"What is the Degree?\">What is the Degree?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/#Partial_Fractions_Decomposition\" title=\"Partial Fractions Decomposition\">Partial Fractions Decomposition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/#Partial_Fractions_Examples_and_Solutions_Integration\" title=\"Partial Fractions Examples and Solutions (Integration)\">Partial Fractions Examples and Solutions (Integration)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/partial-fractions\/#Tips_To_Split_A_Fraction_into_its_Partial_Fractions\" title=\"Tips To Split A Fraction into its Partial Fractions\">Tips To Split A Fraction into its Partial Fractions<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_are_Partial_Fractions\"><\/span>What are Partial Fractions?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A partial fraction is a mathematical expression that is used to represent the quotient of two polynomials. The denominator of the fraction is the polynomial that is divided by the other polynomial. The numerator of the fraction is the polynomial that is left over after the division has been completed.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-155467 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/partial-fractions-types-formulas-examples-and-solutions.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/partial-fractions-types-formulas-examples-and-solutions.jpg?v=1648235442 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/partial-fractions-types-formulas-examples-and-solutions-300x212.jpg?v=1648235442 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Partial_Fractions_from_Rational_Functions\"><\/span>Partial Fractions from Rational Functions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A rational function is a function that can be expressed as the quotient of two polynomials. The denominator of a rational function is always a polynomial. The numerator can be a polynomial, or it can be a rational function.<\/p>\n<p>To find the partial fractions of a rational function, we first need to simplify the function. We can do this by factoring the denominator and cancelling out any common factors.<\/p>\n<p>Once the function is simplified, we can divide the numerator and denominator by the greatest common factor (GCF). This will give us the simplest form of the rational function.<\/p>\n<p>The partial fractions of a rational function are the fractions that make up the simplified function. We can write these fractions in terms of the original function&#8217;s variables.<\/p>\n<p>We can also use the partial fractions to find the inverse of a rational function. To do this, we first need to solve for the roots of the denominator. We can then use these roots to find the inverse of the rational function.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Proper_Fraction_of_Improper_Function\"><\/span>Proper Fraction of Improper Function<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The improper fraction of an improper function is the improper fraction of the function&#8217;s argument.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Heres_How_to_Solve_Partial_Fractions\"><\/span>Here\u2019s How to Solve Partial Fractions!<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To solve partial fractions, we need to decompose the given fraction into a series of simpler fractions.<\/p>\n<p>We can do this by using the algebraic method, or the graphical method.<\/p>\n<p>Algebraic Method<\/p>\n<p>1. Decompose the given fraction into a series of simpler fractions.<\/p>\n<p>2. Write out the algebraic equation for each of the fractions.<\/p>\n<p>3. Solve each of the equations.<\/p>\n<p>4. Combine the solutions to get the final answer.<\/p>\n<p>Graphical Method<\/p>\n<p>1. Draw a graph of the given fraction.<\/p>\n<p>2. Find the points of intersection of the graph and the x-axis.<\/p>\n<p>3. Find the points of intersection of the graph and the y-axis.<\/p>\n<p>4. Write out the algebraic equation for each of the fractions.<\/p>\n<p>5. Solve each of the equations.<\/p>\n<p>6. Combine the solutions to get the final answer.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Now_What_are_Proper_Rational_Expressions\"><\/span>Now, What are Proper Rational Expressions?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A proper rational expression is a rational expression in which the numerator and denominator are polynomials.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_is_the_Degree\"><\/span>What is the Degree?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The degree is a Master of Arts in Teaching.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Partial_Fractions_Decomposition\"><\/span>Partial Fractions Decomposition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\\[\\frac{x}{x^{2}-4}=\\frac{A}{x-2}+\\frac{B}{x-4}+\\frac{C}{x^{2}-4}\\]\n<p>We can rewrite this equation as:<\/p>\n<p>\\[x=A(x-2)+B(x-4)+C(x^{2}-4)\\]\n<p>We can then use the distributive property to simplify this equation:<\/p>\n<p>\\[x=Ax-2+Bx-4+Cx^{2}-4\\]\n<p>We can then combine like terms:<\/p>\n<p>\\[x=Ax-2+Bx-4+Cx^{2}-4=Ax-2+Bx-4-Cx^{2}+4\\]\n<p>We can then move the -Cx^{2} term to the right side of the equation:<\/p>\n<p>\\[x-Cx^{2}=4\\]\n<p>We can then divide both sides of the equation by -Cx^{2}:<\/p>\n<p>\\[x=\\frac{4}{-Cx^{2}}\\]\n<h2><span class=\"ez-toc-section\" id=\"Partial_Fractions_Examples_and_Solutions_Integration\"><\/span>Partial Fractions Examples and Solutions (Integration)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>1) $\\frac{x^2}{4} + \\frac{x}{4}$<\/p>\n<p>We can rewrite the given equation as:<\/p>\n<p>$\\frac{x^2}{4} = \\frac{x}{4} + \\frac{1}{4}$<\/p>\n<p>Now, we can use the standard formula for integrating fractions:<\/p>\n<p>$\\int \\frac{x}{4} dx = \\frac{1}{4}x^2 + C$<\/p>\n<p>2) $\\frac{x^3}{27} + \\frac{x}{9}$<\/p>\n<p>We can rewrite the given equation as:<\/p>\n<p>$\\frac{x^3}{27} = \\frac{x}{9} + \\frac{3}{9}$<\/p>\n<p>Now, we can use the standard formula for integrating fractions:<\/p>\n<p>$\\int \\frac{x}{9} dx = \\frac{3}{9}x^3 + C$<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Tips_To_Split_A_Fraction_into_its_Partial_Fractions\"><\/span>Tips To Split A Fraction into its Partial Fractions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>There are a few steps in order to split a fraction into its partial fractions.<\/p>\n<p>1. Decide which denominator to begin with. In some cases, all of the denominators may be the same.<\/p>\n<p>2. Write the original fraction as a fraction with the chosen denominator.<\/p>\n<p>3. Factor the original fraction&#8217;s denominator.<\/p>\n<p>4. Write each factor of the denominator as a fraction with the chosen denominator.<\/p>\n<p>5. Add the fractions with the chosen denominators.<\/p>\n<p>6. Write the final answer as a fraction with the original denominator.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What are Partial Fractions? 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