{"id":803,"date":"2024-04-29T20:11:25","date_gmt":"2024-04-29T20:11:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=803"},"modified":"2025-02-18T15:26:27","modified_gmt":"2025-02-18T15:26:27","slug":"polynomial-basics-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/polynomial-basics-learn-it-2\/","title":{"raw":"Polynomial Basics: Learn It 2","rendered":"Polynomial Basics: Learn It 2"},"content":{"raw":"<h2>Adding and Subtracting Polynomials<\/h2>\r\n<h3>Add and Subtract Monomials<\/h3>\r\nWhen you are adding or subtracting monomials, think of it as grouping together similar items. These 'like terms' are expressions that have the same variable raised to the same power. It's essential to ensure that both the variable and its exponent match before combining them.\r\n\r\n<section class=\"textbox recall\">When you're bringing like terms together, remember that only the coefficients (the numbers in front of the variables) get added or subtracted. The exponents (the powers to which the variables are raised) always remain unchanged.<\/section><section class=\"textbox example\">Add: [latex]17{x}^{2}+6{x}^{2}[\/latex][reveal-answer q=\"817872\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"817872\"]\r\n<table id=\"eip-id1168469711517\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]17{x}^{2}+6{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]23{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Subtract: [latex]11n-\\left(-8n\\right)[\/latex][reveal-answer q=\"72881\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"72881\"]\r\n<table id=\"eip-id1168468510729\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]11n-\\left(-8n\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]19n[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h3>Add and Subtract Polynomials<\/h3>\r\nAdding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms\u2014those with the same variables with the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together. It may also be helpful to underline, circle, or box like terms.\r\n\r\n<section class=\"textbox example\">Find the sum: [latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex].[reveal-answer q=\"76420\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"76420\"]\r\n<table id=\"eip-id1168468393629\" class=\"unnumbered unstyled\" summary=\"The top row shows parentheses 4 x squared minus 5x plus 1 plus parentheses 3 x squared minus 8x minus 9. The next row says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identify like terms.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224334\/CNX_BMath_Figure_10_01_003-02.png\" alt=\"4x^2-5x+1+3x^2-8x-9 where 4x^2 has a double underline, 5x has a single underline, 1 has a box around it, 3x^2 has a double underline, 8x has a single underline and a 9 has a box around it. \" width=\"285\" height=\"29\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange to get the like terms together.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224335\/CNX_BMath_Figure_10_01_003_img-03.png\" alt=\"4x^2+3x^2-5x-8x+1-9 where 4x^2+3x^2 has a double underline, 5x-8x has a single underline, 1 and 9 area each boxed individually.\" width=\"253\" height=\"30\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]7x^2-13x-8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3863[\/ohm2_question]<\/section><section class=\"textbox example\">Subtract [latex]\\left(9{m}^{2}-7m+4\\right) - \\left({m}^{2}-3m+8\\right)[\/latex][reveal-answer q=\"797347\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"797347\"]\r\n<table id=\"eip-id1168467283477\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](9{m}^{2}-7m+4)-({m}^{2}-3m+8)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute and identify like terms.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224343\/CNX_BMath_Figure_10_01_005_img-02.png\" alt=\"9m^2-7m+4-m^2+3x-8 where 9m^2 has a double underline, 7m has a single underline, 4 has a box around it, m^2 has a double underline, 3m has a single underline, and 8 has a box around it.\" width=\"341\" height=\"28\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the terms.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224345\/CNX_BMath_Figure_10_01_005_img-03.png\" alt=\"9m^2-m^2-7m+3m+4+8 where 9m^2-m^2 has a double underline, 7m+3m has a single underline and 4 and 8 each have their own box around each number.\" width=\"341\" height=\"27\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]8m^2-4m-4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Be careful When subtracting polynomials\r\n[latex]\\\\[\/latex]\r\n<\/strong>When subtracting a polynomial from another, be careful to subtract\u00a0<em>each term<\/em> in the second from the first. That is, use the distributive property to distribute the minus sign through the second polynomial.<center>[latex]\\begin{array}{cc}\\left(3x^2-2x+9\\right)-\\left(x^2-4x+5\\right)\\text{}\\hfill &amp;\\text{Distribute the negative in front of the parenthesis} \\hfill \\\\\r\n3x^2-2x+9 -x^2 -\\left(-4x\\right) - 5\\hfill &amp; \\text{Be careful when subtracting a negative}.\\hfill \\\\ 3x^2 - x^2 -2x+4x+9-5\\hfill &amp; \\text{Rearrange terms in descending order of degree} \\hfill \\\\ 2x^2 +2x +4 \\hfill &amp; \\text{Combine like terms}. \\hfill \\end{array}[\/latex]<\/center><\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3864[\/ohm2_question]<\/section>","rendered":"<h2>Adding and Subtracting Polynomials<\/h2>\n<h3>Add and Subtract Monomials<\/h3>\n<p>When you are adding or subtracting monomials, think of it as grouping together similar items. These &#8216;like terms&#8217; are expressions that have the same variable raised to the same power. It&#8217;s essential to ensure that both the variable and its exponent match before combining them.<\/p>\n<section class=\"textbox recall\">When you&#8217;re bringing like terms together, remember that only the coefficients (the numbers in front of the variables) get added or subtracted. The exponents (the powers to which the variables are raised) always remain unchanged.<\/section>\n<section class=\"textbox example\">Add: [latex]17{x}^{2}+6{x}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q817872\">Show Answer<\/button><\/p>\n<div id=\"q817872\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168469711517\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]17{x}^{2}+6{x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]23{x}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Subtract: [latex]11n-\\left(-8n\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q72881\">Show Answer<\/button><\/p>\n<div id=\"q72881\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168468510729\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]11n-\\left(-8n\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]19n[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<h3>Add and Subtract Polynomials<\/h3>\n<p>Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms\u2014those with the same variables with the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together. It may also be helpful to underline, circle, or box like terms.<\/p>\n<section class=\"textbox example\">Find the sum: [latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q76420\">Show Answer<\/button><\/p>\n<div id=\"q76420\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168468393629\" class=\"unnumbered unstyled\" summary=\"The top row shows parentheses 4 x squared minus 5x plus 1 plus parentheses 3 x squared minus 8x minus 9. The next row says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Identify like terms.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224334\/CNX_BMath_Figure_10_01_003-02.png\" alt=\"4x^2-5x+1+3x^2-8x-9 where 4x^2 has a double underline, 5x has a single underline, 1 has a box around it, 3x^2 has a double underline, 8x has a single underline and a 9 has a box around it.\" width=\"285\" height=\"29\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rearrange to get the like terms together.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224335\/CNX_BMath_Figure_10_01_003_img-03.png\" alt=\"4x^2+3x^2-5x-8x+1-9 where 4x^2+3x^2 has a double underline, 5x-8x has a single underline, 1 and 9 area each boxed individually.\" width=\"253\" height=\"30\" \/><\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]7x^2-13x-8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3863\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3863&theme=lumen&iframe_resize_id=ohm3863&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Subtract [latex]\\left(9{m}^{2}-7m+4\\right) - \\left({m}^{2}-3m+8\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q797347\">Show Answer<\/button><\/p>\n<div id=\"q797347\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168467283477\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex](9{m}^{2}-7m+4)-({m}^{2}-3m+8)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute and identify like terms.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224343\/CNX_BMath_Figure_10_01_005_img-02.png\" alt=\"9m^2-7m+4-m^2+3x-8 where 9m^2 has a double underline, 7m has a single underline, 4 has a box around it, m^2 has a double underline, 3m has a single underline, and 8 has a box around it.\" width=\"341\" height=\"28\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rearrange the terms.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224345\/CNX_BMath_Figure_10_01_005_img-03.png\" alt=\"9m^2-m^2-7m+3m+4+8 where 9m^2-m^2 has a double underline, 7m+3m has a single underline and 4 and 8 each have their own box around each number.\" width=\"341\" height=\"27\" \/><\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]8m^2-4m-4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Be careful When subtracting polynomials<br \/>\n[latex]\\\\[\/latex]<br \/>\n<\/strong>When subtracting a polynomial from another, be careful to subtract\u00a0<em>each term<\/em> in the second from the first. That is, use the distributive property to distribute the minus sign through the second polynomial.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\left(3x^2-2x+9\\right)-\\left(x^2-4x+5\\right)\\text{}\\hfill &\\text{Distribute the negative in front of the parenthesis} \\hfill \\\\  3x^2-2x+9 -x^2 -\\left(-4x\\right) - 5\\hfill & \\text{Be careful when subtracting a negative}.\\hfill \\\\ 3x^2 - x^2 -2x+4x+9-5\\hfill & \\text{Rearrange terms in descending order of degree} \\hfill \\\\ 2x^2 +2x +4 \\hfill & \\text{Combine like terms}. \\hfill \\end{array}[\/latex]<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3864\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3864&theme=lumen&iframe_resize_id=ohm3864&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":5,"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\/803"}],"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\/12"}],"version-history":[{"count":21,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/803\/revisions"}],"predecessor-version":[{"id":7457,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/803\/revisions\/7457"}],"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\/803\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=803"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=803"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=803"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=803"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}