{"id":33741,"date":"2017-10-20T20:33:55","date_gmt":"2017-10-20T20:33:55","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=33741"},"modified":"2026-03-28T10:45:03","modified_gmt":"2026-03-28T15:45:03","slug":"intro-to-polynomials","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/intro-to-polynomials\/","title":{"rendered":"Intro to Polynomials"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_jHjF90C6X4c\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_jHjF90C6X4c\" data-source-videoID=\"jHjF90C6X4c\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Intro to Polynomials Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Intro to Polynomials\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_jHjF90C6X4c:hover {cursor:pointer;} img#videoThumbnailImage_jHjF90C6X4c {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/58-intro-to-polynomials-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_jHjF90C6X4c\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_jHjF90C6X4c\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Intro to Polynomials\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_jHjF90C6X4c\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_jHjF90C6X4c\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_jHjF90C6X4c\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction lHf_Function() {\n  var x = document.getElementById(\"lHf\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"lHf_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"lHf\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Properties_of_Polynomials\" class=\"smooth-scroll\">Properties of Polynomials<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Identifying_Polynomials\" class=\"smooth-scroll\">Identifying Polynomials<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Polynomials_with_Multiple_Terms\" class=\"smooth-scroll\">Polynomials with Multiple Terms<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Writing_Polynomials\" class=\"smooth-scroll\">Writing Polynomials<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Polynomial_Practice_Questions\" class=\"smooth-scroll\">Polynomial Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hey, guys! Welcome to this Mometrix introduction video to polynomials.<\/p>\n<h2><span id=\"Properties_of_Polynomials\" class=\"m-toc-anchor\"><\/span>Properties of Polynomials<\/h2>\n<p>\nPolynomial comes from <em>poly<\/em>, which means &#8220;many,&#8221; and <em>nomial<\/em>, meaning &#8220;term.&#8221; So, really it just says &#8220;many terms.&#8221;<\/p>\n<p>Polynomials can have <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/exponents\/\">exponents<\/a> like the 4 in \\(x^{4}\\), they can have variables like \\(x\\) and \\(y\\), and they can have constants like 7,000, -3 million, \\(\\frac{1}{4}\\), 1, etc.<\/p>\n<p>You can even combine all of these using addition, subtraction, multiplication and division. However, a polynomial can never be divided by a variable, only by a constant.<\/p>\n<h2><span id=\"Identifying_Polynomials\" class=\"m-toc-anchor\"><\/span>Identifying Polynomials<\/h2>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nFor example:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{-28}{7x+1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis is not a polynomial, because there is a variable in the denominator.<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nNow, we have to be careful and remember all of our algebra tricks. Take a look at this.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{-28}{7x^{-2}+1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nBecause the exponent in the bottom has a negative exponent, we know that, when simplified, the exponent is actually moved to the top, which means that this is a polynomial.<\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nLet\u2019s look at another example:<\/p>\n<div class=\"examplesentence\">\\(290yx^{-5}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis is not a polynomial, because when simplified the variable \\(y\\) is moved from the numerator to the denominator, because of the negative exponent; and a polynomial cannot have a variable in the denominator.<\/p>\n<h3><span id=\"Example_4\" class=\"m-toc-anchor\"><\/span>Example #4<\/h3>\n<p>\nNow, try this one:<\/p>\n<div class=\"examplesentence\">\\(7xy^{24}+28x-7y^{23}-3\\)<\/div>\n<p>\n&nbsp;<br \/>\nPolynomial or not?<\/p>\n<p>This one is a polynomial! It does not have a variable in the denominator.<\/p>\n<h2><span id=\"Polynomials_with_Multiple_Terms\" class=\"m-toc-anchor\"><\/span>Polynomials with Multiple Terms<\/h2>\n<p>\nAnother thing that is important to know, is that a polynomial can also just have one term. For instance, 5 alone is a polynomial. This may seem confusing, given that polynomial actually says \u201cmany terms,\u201d so I want to make that clear.<\/p>\n<p>Polynomials with 1, 2, or 3 terms all have special names. The name for a polynomial with 1 term is a monomial, a polynomial with 2 terms is a binomial, and a polynomial with 3 terms, as you might guess, is a trinomial. There is also the quadrinomial with 4 terms, and the quintinomial with 5 terms; but those are not often used.<\/p>\n<p>A polynomial can have as many terms as you need, but it cannot have an infinite number of terms.<\/p>\n<p>Polynomials are really great in math because they are easy to work with. If you add polynomials together, you are still left with a polynomial. If you multiply polynomials together, you still have a polynomial.<\/p>\n<p>You can divide polynomials; however, the result may not be a polynomial.<\/p>\n<p>Also, polynomials with one variable are great for graphing because their lines are always smooth and continuous.<\/p>\n<h2><span id=\"Writing_Polynomials\" class=\"m-toc-anchor\"><\/span>Writing Polynomials<\/h2>\n<p>\nThe last thing you need to know about a polynomial is the standard way to write a polynomial. The standard way to write a polynomial is in descending order, with the highest degree first (farthest to the left), and the lowest degree last. Degrees, in this sense, refer to the number in the exponent.<\/p>\n<p>So, to put the polynomial, \\(7x-7+24x^{3}+1x^{7}\\), you would move the term with the highest number in the exponent to the front. In this case, you would move \\(1x^{7}\\) to the front, and so on. Once you have completed, the standard form of this polynomial is \\(1x^{7}+24x^{3}+7x-7\\).<\/p>\n<p>I hope this video helped you to understand polynomials a little better. See you next time!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Polynomial_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Polynomial Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich expression is NOT a polynomial? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\(\\large{\\frac{18^3-30x^2}{6x}}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(8xy+5\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\">\\(5x^2-\\large{\\frac{8}{x}}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\(\\large{\\frac{1}{2}}\\normalsize{x^2+8x-7}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>A polynomial is an expression that can have many terms with variables, coefficients, and positive exponents. These terms can be added, subtracted, multiplied, and divided.<\/p>\n<p>When simplified, a polynomial can\u2019t have a variable in its denominator, and it can\u2019t contain a negative exponent. The expression \\(5x^2-8x\\) has the variable \\(x\\), in its denominator, so this is not a polynomial.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich statement is true? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">A polynomial can have an infinite number of terms.<\/div><div class=\"PQ\"  id=\"PQ-2-2\">A polynomial can contain a negative exponent. <\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">An expression with one term can be a polynomial. <\/div><div class=\"PQ\"  id=\"PQ-2-4\">A polynomial with two terms is called a trinomial. <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Although the word <em>polynomial<\/em> means &#8220;multiple terms,&#8221; a polynomial can also have exactly one term. A polynomial with one term is called a monomial.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich statement is false? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">If you add two polynomial expressions, the result is a polynomial. <\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">If you divide two polynomial expressions, the result is a polynomial. <\/div><div class=\"PQ\"  id=\"PQ-3-3\">If you multiply two polynomial expressions, the result is a polynomial.<\/div><div class=\"PQ\"  id=\"PQ-3-4\">Polynomials with one variable graph as a smooth and continuous line. <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>When dividing polynomials, the result may not be a polynomial.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe length of a rectangle is written as \\(8x+5+x^2\\). Its width is written as \\(4+6x\\). Which answer option shows both polynomials written in standard form?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-4-1\">Length: \\(x^2+8x+5\\)<br>\r\nWidth: \\(6x+4\\)<\/div><div class=\"PQ\"  id=\"PQ-4-2\">Length: \\(5+8x+x^2\\)<br>\r\nWidth: \\(4+6x\\)<\/div><div class=\"PQ\"  id=\"PQ-4-3\">Length: \\(8x+x^2+5\\)<br>\r\nWidth: \\(4x+6\\)<\/div><div class=\"PQ\"  id=\"PQ-4-4\">Length: \\(x^2-8x-5\\)<br>\r\nWidth: \\(6+4x\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Polynomials are written in descending order, with the highest degree first and the lowest degree last. Since the highest degree in the length is 2, the term \\(x^2\\) must come first in the expression. This is followed by \\(8x\\), and 5 is written last.<\/p>\n<p>Since the highest degree in the width is 1, the term \\(6x\\) must come first in the expression, followed by 4.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nAllison and Christina are racing to see who can run faster around a track. Since Christina had more time to practice, she agreed to give Allison a head start of 50 meters. After \\(x\\) seconds, Allison is \\(8x+50\\) meters from the starting line. What type of polynomial is Allison\u2019s distance from the starting line, \\(8x+50\\)? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">Quadrinomial<\/div><div class=\"PQ\"  id=\"PQ-5-2\">Trinomial<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-3\">Binomial<\/div><div class=\"PQ\"  id=\"PQ-5-4\">Monomial<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>A binomial is a polynomial that contains two terms. In the expression \\(8x+50\\), the two terms are \\(8x\\) and 50.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-i\/\">Return to Algebra I Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra I Videos<\/p>\n","protected":false},"author":1,"featured_media":90994,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-33741","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-algebra-basics-videos","7":"page_category-math-advertising-group","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/33741","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=33741"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/33741\/revisions"}],"predecessor-version":[{"id":280565,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/33741\/revisions\/280565"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/90994"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=33741"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}