{"id":2605,"date":"2025-08-13T18:10:47","date_gmt":"2025-08-13T18:10:47","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2605"},"modified":"2026-01-14T18:05:12","modified_gmt":"2026-01-14T18:05:12","slug":"rational-functions-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-background-youll-need-3\/","title":{"raw":"Rational Functions: Background You'll Need 3","rendered":"Rational Functions: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Simplify fractions using factors<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>simplifying fractions<\/h3>\r\nSimplifying fractions involves finding common factors in the numerator and denominator, then canceling them out.\r\n\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">From your work with basic fractions, you might remember that [latex]\\frac{12}{18} = \\frac{2 \\cdot 6}{3 \\cdot 6} = \\frac{2}{3}[\/latex] by canceling the common factor of 6. The same principle applies to algebraic expressions.<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Simplify [latex]\\frac{(x-3)(x+2)}{(x-3)(x+5)}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned}= \\frac{(x-3)(x+2)}{(x-3)(x+5)} \\quad \\text{identify common factor } (x-3)\\\\ &amp;= \\frac{x+2}{x+5} \\quad \\text{cancel the common factor} \\end{aligned}[\/latex]<\/p>\r\n\r\n<\/section>Most rational expressions won't be in factored form so it's important we review how to factor as well.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">There are multiple factoring techniques:\r\n<ul>\r\n \t<li>GCF:\r\n<p class=\"whitespace-normal break-words\">Find the GCF of [latex]12x^3y^2[\/latex] and [latex]18x^2y^4[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} 12x^3y^2 &amp;= 2^2 \\cdot 3 \\cdot x^3 \\cdot y^2\\ 18x^2y^4 \\\\&amp;= 2 \\cdot 3^2 \\cdot x^2 \\cdot y^4\\ \\text{GCF} \\\\&amp;= 2 \\cdot 3 \\cdot x^2 \\cdot y^2 = 6x^2y^2 \\end{aligned}[\/latex]<\/p>\r\n<\/li>\r\n \t<li>Trinomial:\r\n<p class=\"whitespace-normal break-words\">Factor [latex]x^2 - 9x + 18[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]= (x - 3)(x - 6) \\quad \\text{find two numbers that multiply to 18 and add to -9} \\end{aligned}[\/latex]<\/p>\r\n<\/li>\r\n \t<li>Difference of squares:\r\n<p class=\"whitespace-normal break-words\">Factor [latex]4x^2 - 25[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} 4x^2 - 25 = \\&amp; (2x)^2 - 5^2 \\quad \\text{recognize difference of squares}\\\\ &amp;= (2x + 5)(2x - 5) \\end{aligned}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Simplify [latex]\\frac{x^2 - 4}{x^2 + 4x + 4}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} \\frac{x^2 - 4}{x^2 + 4x + 4} &amp; = \\frac{(x-2)(x+2)}{(x+2)^2} \\quad \\text{factor both numerator and denominator} \\\\ &amp;= \\frac{(x-2)(x+2)}{(x+2)(x+2)} \\quad \\text{identify common factor}\\\\ &amp;= \\frac{x-2}{x+2} \\quad \\text{cancel one factor of } (x+2) \\end{aligned}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">You can only cancel common factors, never common terms. For instance, in [latex]\\frac{x + 3}{x + 5}[\/latex], you cannot cancel the [latex]x[\/latex] terms because they are being added, not multiplied.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311151[\/ohm_question]<\/section>\r\n<p class=\"whitespace-normal break-words\">Now you're ready to work with more complex rational expressions and understand how rational functions behave.<\/p>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Simplify fractions using factors<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>simplifying fractions<\/h3>\n<p>Simplifying fractions involves finding common factors in the numerator and denominator, then canceling them out.<\/p>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">From your work with basic fractions, you might remember that [latex]\\frac{12}{18} = \\frac{2 \\cdot 6}{3 \\cdot 6} = \\frac{2}{3}[\/latex] by canceling the common factor of 6. The same principle applies to algebraic expressions.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Simplify [latex]\\frac{(x-3)(x+2)}{(x-3)(x+5)}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned}= \\frac{(x-3)(x+2)}{(x-3)(x+5)} \\quad \\text{identify common factor } (x-3)\\\\ &= \\frac{x+2}{x+5} \\quad \\text{cancel the common factor} \\end{aligned}[\/latex]<\/p>\n<\/section>\n<p>Most rational expressions won&#8217;t be in factored form so it&#8217;s important we review how to factor as well.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">There are multiple factoring techniques:<\/p>\n<ul>\n<li>GCF:\n<p class=\"whitespace-normal break-words\">Find the GCF of [latex]12x^3y^2[\/latex] and [latex]18x^2y^4[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} 12x^3y^2 &= 2^2 \\cdot 3 \\cdot x^3 \\cdot y^2\\ 18x^2y^4 \\\\&= 2 \\cdot 3^2 \\cdot x^2 \\cdot y^4\\ \\text{GCF} \\\\&= 2 \\cdot 3 \\cdot x^2 \\cdot y^2 = 6x^2y^2 \\end{aligned}[\/latex]<\/p>\n<\/li>\n<li>Trinomial:\n<p class=\"whitespace-normal break-words\">Factor [latex]x^2 - 9x + 18[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]= (x - 3)(x - 6) \\quad \\text{find two numbers that multiply to 18 and add to -9} \\end{aligned}[\/latex]<\/p>\n<\/li>\n<li>Difference of squares:\n<p class=\"whitespace-normal break-words\">Factor [latex]4x^2 - 25[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} 4x^2 - 25 = \\& (2x)^2 - 5^2 \\quad \\text{recognize difference of squares}\\\\ &= (2x + 5)(2x - 5) \\end{aligned}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Simplify [latex]\\frac{x^2 - 4}{x^2 + 4x + 4}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{aligned} \\frac{x^2 - 4}{x^2 + 4x + 4} & = \\frac{(x-2)(x+2)}{(x+2)^2} \\quad \\text{factor both numerator and denominator} \\\\ &= \\frac{(x-2)(x+2)}{(x+2)(x+2)} \\quad \\text{identify common factor}\\\\ &= \\frac{x-2}{x+2} \\quad \\text{cancel one factor of } (x+2) \\end{aligned}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">You can only cancel common factors, never common terms. For instance, in [latex]\\frac{x + 3}{x + 5}[\/latex], you cannot cancel the [latex]x[\/latex] terms because they are being added, not multiplied.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311151\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311151&theme=lumen&iframe_resize_id=ohm311151&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"whitespace-normal break-words\">Now you&#8217;re ready to work with more complex rational expressions and understand how rational functions behave.<\/p>\n","protected":false},"author":67,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":508,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2605"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2605\/revisions"}],"predecessor-version":[{"id":3756,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2605\/revisions\/3756"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/508"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2605\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2605"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2605"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2605"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2605"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}