{"id":3039,"date":"2024-08-28T15:28:31","date_gmt":"2024-08-28T15:28:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=3039"},"modified":"2024-11-20T02:40:31","modified_gmt":"2024-11-20T02:40:31","slug":"polynomial-basics-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/polynomial-basics-learn-it-5\/","title":{"raw":"Polynomial Basics: Learn It 5","rendered":"Polynomial Basics: Learn It 5"},"content":{"raw":"<h2>Performing Operations with Polynomials of Several Variables<\/h2>\r\nWe have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\left(a+2b\\right)\\left(4a-b-c\\right)\\hfill &amp; \\hfill \\\\ a\\left(4a-b-c\\right)+2b\\left(4a-b-c\\right)\\hfill &amp; \\text{Use the distributive property}.\\hfill \\\\ 4{a}^{2}-ab-ac+8ab - 2{b}^{2}-2bc\\hfill &amp; \\text{Multiply}.\\hfill \\\\ 4{a}^{2}+\\left(-ab+8ab\\right)-ac - 2{b}^{2}-2bc\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 4{a}^{2}+7ab-ac - 2bc - 2{b}^{2}\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\r\n<section class=\"textbox example\">Multiply [latex]\\left(x+4\\right)\\left(3x - 2y+5\\right)[\/latex].Follow the same steps that we used to multiply polynomials containing only one variable.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}x\\left(3x - 2y+5\\right)+4\\left(3x - 2y+5\\right) \\hfill &amp; \\text{Use the distributive property}.\\hfill \\\\ 3{x}^{2}-2xy+5x+12x - 8y+20\\hfill &amp; \\text{Multiply}.\\hfill \\\\ 3{x}^{2}-2xy+\\left(5x+12x\\right)-8y+20\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 3{x}^{2}-2xy+17x - 8y+20 \\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18878[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18877[\/ohm2_question]<\/section>","rendered":"<h2>Performing Operations with Polynomials of Several Variables<\/h2>\n<p>We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\left(a+2b\\right)\\left(4a-b-c\\right)\\hfill & \\hfill \\\\ a\\left(4a-b-c\\right)+2b\\left(4a-b-c\\right)\\hfill & \\text{Use the distributive property}.\\hfill \\\\ 4{a}^{2}-ab-ac+8ab - 2{b}^{2}-2bc\\hfill & \\text{Multiply}.\\hfill \\\\ 4{a}^{2}+\\left(-ab+8ab\\right)-ac - 2{b}^{2}-2bc\\hfill & \\text{Combine like terms}.\\hfill \\\\ 4{a}^{2}+7ab-ac - 2bc - 2{b}^{2}\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<section class=\"textbox example\">Multiply [latex]\\left(x+4\\right)\\left(3x - 2y+5\\right)[\/latex].Follow the same steps that we used to multiply polynomials containing only one variable.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}x\\left(3x - 2y+5\\right)+4\\left(3x - 2y+5\\right) \\hfill & \\text{Use the distributive property}.\\hfill \\\\ 3{x}^{2}-2xy+5x+12x - 8y+20\\hfill & \\text{Multiply}.\\hfill \\\\ 3{x}^{2}-2xy+\\left(5x+12x\\right)-8y+20\\hfill & \\text{Combine like terms}.\\hfill \\\\ 3{x}^{2}-2xy+17x - 8y+20 \\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18878\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18878&theme=lumen&iframe_resize_id=ohm18878&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18877\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18877&theme=lumen&iframe_resize_id=ohm18877&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":55,"module-header":"learn_it","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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3039"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3039\/revisions"}],"predecessor-version":[{"id":3046,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3039\/revisions\/3046"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/55"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3039\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=3039"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=3039"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=3039"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=3039"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}