{"id":1680,"date":"2024-04-24T14:57:01","date_gmt":"2024-04-24T14:57:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1680"},"modified":"2025-10-31T13:50:36","modified_gmt":"2025-10-31T13:50:36","slug":"introduction-to-derivatives-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/introduction-to-derivatives-background-youll-need-1\/","title":{"raw":"Introduction to Derivatives: Background You'll Need 1","rendered":"Introduction to Derivatives: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Add and subtract polynomials<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Adding and Subtracting Polynomials<\/h2>\r\n<p>The process of adding and subtracting polynomials is a fundamental skill in algebra that involves combining like terms. Like terms are terms that have exactly the same variable factors raised to the same powers, or exponents.<\/p>\r\n<p>When working with polynomials, it's essential to identify like terms because these are the only terms that can be combined through addition or subtraction.<\/p>\r\n<section class=\"textbox example\">\r\n<p>[latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms because they both contain the variable [latex]x[\/latex] raised to the second power. Thus, combining them is straightforward: [latex]3{x}^{2}[\/latex].<br \/>\r\n<br \/>\r\nIn contrast [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms and therefore cannot be added. are not like terms since their exponents differ ([latex]x^1[\/latex] vs. [latex]x^2[\/latex]). They cannot be combined through addition or subtraction.<\/p>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Combining Polynomials using Addition of Subtraction<br \/>\r\n<\/strong><\/p>\r\n<ol>\r\n\t<li><strong>Identify and Group Like Terms<\/strong>: Begin by scanning the polynomial expression for terms that have the same variables raised to the same powers. Group these terms together for simplification.<\/li>\r\n\t<li><strong>Combine Like Terms<\/strong>: Add or subtract the coefficients (numerical parts) of like terms. The variable part will remain unchanged.<\/li>\r\n\t<li><strong>Simplify and Write in Standard Form<\/strong>: Once the like terms are combined, ensure that the polynomial is written in standard form.\u00a0<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<p>The standard form of a polynomial starts with the term that has the highest power (degree) and proceeds in descending order of degree.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Add the following polynomials:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(12{x}^{2}+9x - 21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\r\n<p>[reveal-answer q=\"660613\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"660613\"]<\/p>\r\n<center>[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/center>\r\n<p>We can check our answers to these types of problems using a graphing calculator.<\/p>\r\n<p>To check, graph the original problem as given along with the simplified answer. The two graphs should be the same.<\/p>\r\n<p>Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_number=1]288283[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p><strong>Be careful When subtracting polynomials<\/strong><\/p>\r\n<p>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>\r\n<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 \\\\<br \/>\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>\r\n<section class=\"textbox example\">\r\n<p>Subtract the following polynomials:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\r\n<p>[reveal-answer q=\"831247\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"831247\"]<\/p>\r\n<center>[latex]\\begin{array}{cc}7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/center>\r\n<p>Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_number=1]288284[\/ohm_question]<\/p>\r\n<\/section>\r\n<p>Watch this video to see more examples of adding and subtracting polynomials.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>https:\/\/youtu.be\/jiq3toC7wGM<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Add and subtract polynomials<\/li>\n<\/ul>\n<\/section>\n<h2>Adding and Subtracting Polynomials<\/h2>\n<p>The process of adding and subtracting polynomials is a fundamental skill in algebra that involves combining like terms. Like terms are terms that have exactly the same variable factors raised to the same powers, or exponents.<\/p>\n<p>When working with polynomials, it&#8217;s essential to identify like terms because these are the only terms that can be combined through addition or subtraction.<\/p>\n<section class=\"textbox example\">\n<p>[latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms because they both contain the variable [latex]x[\/latex] raised to the second power. Thus, combining them is straightforward: [latex]3{x}^{2}[\/latex].<\/p>\n<p>In contrast [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms and therefore cannot be added. are not like terms since their exponents differ ([latex]x^1[\/latex] vs. [latex]x^2[\/latex]). They cannot be combined through addition or subtraction.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Combining Polynomials using Addition of Subtraction<br \/>\n<\/strong><\/p>\n<ol>\n<li><strong>Identify and Group Like Terms<\/strong>: Begin by scanning the polynomial expression for terms that have the same variables raised to the same powers. Group these terms together for simplification.<\/li>\n<li><strong>Combine Like Terms<\/strong>: Add or subtract the coefficients (numerical parts) of like terms. The variable part will remain unchanged.<\/li>\n<li><strong>Simplify and Write in Standard Form<\/strong>: Once the like terms are combined, ensure that the polynomial is written in standard form.\u00a0<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox recall\">\n<p>The standard form of a polynomial starts with the term that has the highest power (degree) and proceeds in descending order of degree.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Add the following polynomials:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(12{x}^{2}+9x - 21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q660613\">Show Solution<\/button><\/p>\n<div id=\"q660613\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill & \\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x - 1\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<p>We can check our answers to these types of problems using a graphing calculator.<\/p>\n<p>To check, graph the original problem as given along with the simplified answer. The two graphs should be the same.<\/p>\n<p>Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288283\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288283&theme=lumen&iframe_resize_id=ohm288283&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox proTip\">\n<p><strong>Be careful When subtracting polynomials<\/strong><\/p>\n<p>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 \\\\<br \/>  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 example\">\n<p>Subtract the following polynomials:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q831247\">Show Solution<\/button><\/p>\n<div id=\"q831247\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill & \\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<p>Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288284\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288284&theme=lumen&iframe_resize_id=ohm288284&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>Watch this video to see more examples of adding and subtracting polynomials.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Adding and Subtracting Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jiq3toC7wGM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"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":503,"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1680"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":22,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1680\/revisions"}],"predecessor-version":[{"id":4851,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1680\/revisions\/4851"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/503"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1680\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1680"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1680"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1680"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1680"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}