{"id":2693,"date":"2025-08-13T18:27:57","date_gmt":"2025-08-13T18:27:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2693"},"modified":"2025-09-25T20:36:40","modified_gmt":"2025-09-25T20:36:40","slug":"probability-and-counting-theory-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/probability-and-counting-theory-background-youll-need-1\/","title":{"raw":"Probability and Counting Theory: Background You'll Need 1","rendered":"Probability and Counting Theory: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Expand squared and cubed binomials<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Expanding Squared Binomials<\/h2>\r\n<p class=\"whitespace-normal break-words\">When we square a binomial like [latex](a + b)^2[\/latex], we're multiplying [latex](a + b)(a + b)[\/latex].<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Expand [latex](x + 4)^2[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">\r\n[reveal-answer q=\"397282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"397282\"][latex] \\begin{align} (x + 4)^2 &amp;= (x + 4)(x + 4) \\ &amp;= x \\cdot x + x \\cdot 4 + 4 \\cdot x + 4 \\cdot 4 \\quad \\text{use FOIL} \\ &amp;= x^2 + 4x + 4x + 16 \\ &amp;= x^2 + 8x + 16 \\quad \\text{combine like terms} \\end{align} [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Perfect square trinomials glossary: expressions like [latex]x^2 + 8x + 16[\/latex] that result from squaring binomials follow the pattern [latex](a + b)^2 = a^2 + 2ab + b^2[\/latex]. This pattern can save time!\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312747[\/ohm_question]\r\n\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Expanding Cubed Binomials<\/h2>\r\n<p class=\"whitespace-normal break-words\">When we cube a binomial like [latex](a + b)^3[\/latex], we multiply [latex](a + b)(a + b)(a + b)[\/latex].<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex](x + 2)^3[\/latex].\r\n\r\n[reveal-answer q=\"725551\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"725551\"]\r\n<p class=\"whitespace-pre-wrap break-words\">First, let's find [latex](x + 2)^2[\/latex]: [latex] \\begin{align} (x + 2)^2 &amp;= x^2 + 4x + 4 \\end{align} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Now multiply this result by [latex](x + 2)[\/latex]: [latex] \\begin{align} (x + 2)^3 &amp;= (x + 2)^2 \\cdot (x + 2) \\ &amp;= (x^2 + 4x + 4)(x + 2) \\ &amp;= x^2(x + 2) + 4x(x + 2) + 4(x + 2) \\quad \\text{distribute each term} \\ &amp;= x^3 + 2x^2 + 4x^2 + 8x + 4x + 8 \\ &amp;= x^3 + 6x^2 + 12x + 8 \\quad \\text{combine like terms} \\end{align} [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312748[\/ohm_question]\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Expand squared and cubed binomials<\/span><\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Expanding Squared Binomials<\/h2>\n<p class=\"whitespace-normal break-words\">When we square a binomial like [latex](a + b)^2[\/latex], we&#8217;re multiplying [latex](a + b)(a + b)[\/latex].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Expand [latex](x + 4)^2[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q397282\">Show Solution<\/button><\/p>\n<div id=\"q397282\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{align} (x + 4)^2 &= (x + 4)(x + 4) \\ &= x \\cdot x + x \\cdot 4 + 4 \\cdot x + 4 \\cdot 4 \\quad \\text{use FOIL} \\ &= x^2 + 4x + 4x + 16 \\ &= x^2 + 8x + 16 \\quad \\text{combine like terms} \\end{align}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Perfect square trinomials glossary: expressions like [latex]x^2 + 8x + 16[\/latex] that result from squaring binomials follow the pattern [latex](a + b)^2 = a^2 + 2ab + b^2[\/latex]. This pattern can save time!<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312747\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312747&theme=lumen&iframe_resize_id=ohm312747&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Expanding Cubed Binomials<\/h2>\n<p class=\"whitespace-normal break-words\">When we cube a binomial like [latex](a + b)^3[\/latex], we multiply [latex](a + b)(a + b)(a + b)[\/latex].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex](x + 2)^3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q725551\">Show Solution<\/button><\/p>\n<div id=\"q725551\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">First, let&#8217;s find [latex](x + 2)^2[\/latex]: [latex]\\begin{align} (x + 2)^2 &= x^2 + 4x + 4 \\end{align}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Now multiply this result by [latex](x + 2)[\/latex]: [latex]\\begin{align} (x + 2)^3 &= (x + 2)^2 \\cdot (x + 2) \\ &= (x^2 + 4x + 4)(x + 2) \\ &= x^2(x + 2) + 4x(x + 2) + 4(x + 2) \\quad \\text{distribute each term} \\ &= x^3 + 2x^2 + 4x^2 + 8x + 4x + 8 \\ &= x^3 + 6x^2 + 12x + 8 \\quad \\text{combine like terms} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312748\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312748&theme=lumen&iframe_resize_id=ohm312748&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":513,"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\/2693"}],"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":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2693\/revisions"}],"predecessor-version":[{"id":4437,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2693\/revisions\/4437"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/513"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2693\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2693"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2693"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2693"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2693"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}