{"id":207,"date":"2023-01-30T16:28:42","date_gmt":"2023-01-30T16:28:42","guid":{"rendered":"https:\/\/cwassignments.com\/blog\/?p=207"},"modified":"2024-05-10T07:52:39","modified_gmt":"2024-05-10T07:52:39","slug":"simplifying-rational-expressions","status":"publish","type":"post","link":"https:\/\/cwassignments.com\/blog\/simplifying-rational-expressions\/","title":{"rendered":"Simplifying Rational Expressions"},"content":{"rendered":"<h3>Canceling Like Factors<\/h3>\n<p style=\"font-weight: 400;\">When we reduce a common fraction such as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-339\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image002-4.gif\" alt=\"\" width=\"49\" height=\"41\" \/><\/p>\n<p style=\"font-weight: 400;\">we do so by noticing that there is a factor common to both the numerator and the denominator (a factor of 2 in this example), which we can divide out of both the numerator and the denominator.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-341\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image004-3.gif\" alt=\"\" width=\"97\" height=\"48\" \/><\/p>\n<p style=\"font-weight: 400;\">We use exactly the same procedure to reduce rational expressions.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-343\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image006-3.gif\" alt=\"\" width=\"105\" height=\"132\" \/><\/p>\n<h4>Polynomial \/ Monomial<\/h4>\n<p style=\"font-weight: 400;\"><u>Each term<\/u>\u00a0in the numerator must have a factor that cancels a common factor in the denominator.<\/p>\n<p style=\"font-weight: 400;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-345\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image008-3.gif\" alt=\"\" width=\"105\" height=\"44\" \/><\/p>\n<p style=\"font-weight: 400;\">but<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-347\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image010-3.gif\" alt=\"\" width=\"44\" height=\"41\" \/><\/p>\n<p style=\"font-weight: 400;\">cannot be reduced because the 2 is not a common factor of the entire numerator.<\/p>\n<p style=\"font-weight: 400;\"><b><strong>WARNING\u00a0\u00a0<\/strong><\/b>You can only cancel a factor of the\u00a0<u>entire<\/u>\u00a0numerator with a factor of the\u00a0<u>entire<\/u>\u00a0denominator<\/p>\n<table style=\"font-weight: 400;\">\n<tbody>\n<tr>\n<td width=\"590\">However, as an alternative, a fraction with more than one term in the numerator can be split up into separate fractions with each term over the same denominator; then each separate fraction can bereduced if possible:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-349\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image012-2.gif\" alt=\"\" width=\"107\" height=\"127\" \/><\/p>\n<p>\u00b7\u00a0Think of this as the reverse of adding fractions over a common denominator.<\/p>\n<p>Sometimes this is a useful thing to do, depending on the circumstances.You end up with simpler fractions, but the price you pay is that you have more fractions than you started with.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li style=\"font-weight: 400;\">Polynomials must be factored first. You can\u2019t cancel factors unless you can see the factors:<\/li>\n<\/ul>\n<p style=\"font-weight: 400;\"><b><strong>Example:<\/strong><\/b><\/p>\n<p style=\"font-weight: 400;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-351\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image014-2.gif\" alt=\"\" width=\"183\" height=\"157\" \/><\/p>\n<ul>\n<li style=\"font-weight: 400;\">Notice how canceling the (<em>x<\/em>\u2013 2) from the denominator left behind a factor of 1<\/li>\n<\/ul>\n<h3>Multiplication and Division<\/h3>\n<p style=\"font-weight: 400;\">Same rules as for rational numbers!<\/p>\n<h4>Multiplication<\/h4>\n<ul style=\"font-weight: 400;\">\n<li>Both the numerators and the denominators multiply together<\/li>\n<li>Common factors may be cancelled before multiplying<\/li>\n<\/ul>\n<p style=\"font-weight: 400;\"><b><strong>Example:<\/strong><\/b><\/p>\n<table style=\"font-weight: 400;\">\n<tbody>\n<tr>\n<td>Given Equation:<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-353 alignright\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image016-2.gif\" alt=\"\" width=\"123\" height=\"44\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>First factor all the expressions:<br \/>\n(I also put the denominators in parentheses because then it is easier t\u00a0 see them as distinct factors)<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-355 aligncenter\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image018-2.gif\" alt=\"\" width=\"193\" height=\"44\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Now cancel common factors\u2014any factor on the top can cancel with any factor on the bottom:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-357\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image020-1.gif\" alt=\"\" width=\"199\" height=\"53\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-359\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image022-1.gif\" alt=\"\" width=\"165\" height=\"53\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-361\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image024-1.gif\" alt=\"\" width=\"113\" height=\"41\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Now just multiply what\u2019s left.<br \/>\nYou usually do not have to multiply outthe factors, just leave them as shown.<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-365 alignright\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image026.gif\" alt=\"\" width=\"100\" height=\"48\" \/><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Division<\/h4>\n<ul style=\"font-weight: 400;\">\n<li>Multiply by the reciprocal of the divisor<\/li>\n<li>Invert the second fraction, then proceed with multiplication as above<\/li>\n<li>Do not attempt to cancel factors before it is written as a multiplication<\/li>\n<\/ul>\n<h3>Addition and Subtraction<\/h3>\n<p style=\"font-weight: 400;\">Same procedure as for rational numbers!<\/p>\n<ul>\n<li style=\"font-weight: 400;\">Only the numerators can be addedtogether, and only when all the denominators are the same<\/li>\n<\/ul>\n<h4>Finding the LCD<\/h4>\n<ul style=\"font-weight: 400;\">\n<li>The LCD is built up of all the factors of the individual denominators, each factor included the most number of times it appears in an individual denominator.<\/li>\n<li>The product of all the denominators is always a commondenominator, but not necessarily the LCD (the final answer may have to be reduced).<\/li>\n<\/ul>\n<p style=\"font-weight: 400;\"><b><strong>Example:<\/strong><\/b><\/p>\n<table style=\"font-weight: 400;\">\n<tbody>\n<tr>\n<td>Given equation:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-371\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image028.gif\" alt=\"\" width=\"125\" height=\"42\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Factor both denominators:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-373\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image030.gif\" alt=\"\" width=\"177\" height=\"44\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Assemble the LCD:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-375\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image032.gif\" alt=\"\" width=\"168\" height=\"21\" \/><br \/>\nNote that the LCD contains bothdenominators <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-377\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image034.gif\" alt=\"\" width=\"171\" height=\"45\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-381\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image036.gif\" alt=\"\" width=\"171\" height=\"45\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Build up the fractions so that they<br \/>\nboth have the LCD for a denominator: (keep both denominators in factored form to make it easier to see what factors they need to look like the LCD)<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-383\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image038-1.gif\" alt=\"\" width=\"177\" height=\"44\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-385\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image040.gif\" alt=\"\" width=\"266\" height=\"44\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Now that they are over the same denominator, you can add the numerators:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-387\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image042.gif\" alt=\"\" width=\"166\" height=\"44\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>And simplify:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-389\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image044.gif\" alt=\"\" width=\"153\" height=\"47\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-391\" src=\"https:\/\/cwassignments.com\/blog\/wp-content\/uploads\/2023\/01\/image046.gif\" alt=\"\" width=\"147\" height=\"47\" \/><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Confused by Simplifying Rational Expressions in math? Get <a href=\"https:\/\/cwassignments.com\/\">professional assignment help<\/a> to clarify your understanding and excel!<\/p>\n<p>If you need <a href=\"https:\/\/cwassignments.com\/do-my-math-homework.html\">someone to do your math homework<\/a> , you can turn to CWassignments.com.\u00a0 Our experts can help you with all kinds of STEM assignments including &#8216;<a href=\"https:\/\/cwassignments.com\">do my programming homework<\/a>&#8216; help.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Canceling Like Factors When we reduce a common fraction such as we do so by noticing that there is a factor common to both the numerator and the denominator (a factor of 2 in this example), which we can divide out of both the numerator and the denominator. We use exactly the same procedure to [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":702,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"_swt_meta_header_display":false,"_swt_meta_footer_display":false,"_swt_meta_site_title_display":false,"_swt_meta_sticky_header":false,"_swt_meta_transparent_header":false,"footnotes":""},"categories":[7],"tags":[],"class_list":["post-207","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/posts\/207","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/comments?post=207"}],"version-history":[{"count":10,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/posts\/207\/revisions"}],"predecessor-version":[{"id":524,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/posts\/207\/revisions\/524"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/media\/702"}],"wp:attachment":[{"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/media?parent=207"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/categories?post=207"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/tags?post=207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}