{"id":121156,"date":"2022-05-05T10:31:08","date_gmt":"2022-05-05T15:31:08","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=121156"},"modified":"2026-03-28T11:51:33","modified_gmt":"2026-03-28T16:51:33","slug":"multiplying-polynomials-by-monomials","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/multiplying-polynomials-by-monomials\/","title":{"rendered":"Multiplying Polynomials by Monomials"},"content":{"rendered":"<h1>Multiplying Polynomials by Monomials<\/h1>\n\n\t\t\t<div id=\"mmDeferVideoEncompass_qdLgSvabN5g\" 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_qdLgSvabN5g\" data-source-videoID=\"qdLgSvabN5g\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Multiplying Polynomials by Monomials Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Multiplying Polynomials by Monomials\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_qdLgSvabN5g:hover {cursor:pointer;} img#videoThumbnailImage_qdLgSvabN5g {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/2340-thumbnail-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_qdLgSvabN5g\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_qdLgSvabN5g\" 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\",\"Multiplying Polynomials by Monomials\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_qdLgSvabN5g\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_qdLgSvabN5g\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_qdLgSvabN5g\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello! Today we\u2019re going to take a look at how to multiply a <a href=\"https:\/\/www.mometrix.com\/academy\/polynomials\/\"><strong>polynomial<\/strong><\/a> by a monomial. Let\u2019s start with a simple example.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(3x(4x+2)\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>When multiplying a polynomial by a monomial, you&#8217;ll want to multiply each part of the polynomial \\((4x+2)\\) by each part of the monomial \\((3x)\\). You&#8217;ll multiply the monomial, \\(3x\\), by the first part of the polynomial, so by this \\(4x\\). And then we&#8217;ll add \u2014 multiplying the monomial by this part of the polynomial, the \\(+2\\).<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\((3x)(4x)+(3x)(2)\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>So now all we have to do is multiply each of these sets of monomials. <\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(12x^{2}+6x\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>And that&#8217;s our answer. Not too challenging! Let\u2019s try another one.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(7y^{2}(2y^{2}+6y-9)\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Remember, multiply each part of the polynomial by the monomial.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\((7y^{2})(2y^{2})+(7y^{2})(6y)\\)\\(+(7y^{2})(-9)\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Notice that I put plus signs in between each set of multiplied terms, even though there is a \u20139. I did this because subtraction is always the same thing as adding a negative number. Our last sign will end up turning into a subtraction symbol, but doing it this way helps us make sure we have the correct sign attached to each term. <\/p>\n<p>So now let&#8217;s multiply all of our terms. \\(7y^{2}\\cdot 2y^{2}=14y^{4}\\). Remember, when you multiply exponential terms with the same base you simply add the exponents, so we&#8217;ve got \\(2+2=4\\) \u2014 so that&#8217;s where the \\(y^{4}\\) came from. And then, \\(7y^{2}\\cdot 6y=42y^{3}\\), so +42y3. And then, plus \\(7y^{2}\\cdot (-9)=-63y^{2}\\).<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(14y^{4}+42y^{3}+(-63y^{2})\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Remember what I said earlier? Adding a negative number is the same as subtracting, so we can rewrite this as:<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(14y^{4}+42y^{3}-63y^{2}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Let\u2019s try one last problem before we go.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(-5x^{2}y(3x-5xy+10y^{2})\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>We need to multiply each term of the polynomial by the monomial.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\((-5x^{2}y)(3x)\\)\\(+(-5x^{2}y)(-5xy)\\)\\(+(-5x^{2}y)(10y^{2})\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Now we multiply each set of monomials. Remember, a negative number times a negative number is a positive number.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:85%; text-align:center;\">\\(-15x^{3}y+25x^{3}y^{2}+(-50x^{2}y^{3})\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>We can simplify this last sign, and our final answer will be:<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(-15x^{3}y+25x^{3}y^{2}-50x^{2}y^{3}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>\nI hope that this video on multiplying polynomials by monomials has been helpful! Thanks for watching, and happy studying!<\/p>\n<\/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>Multiplying Polynomials by Monomials Return to Algebra I Videos<\/p>\n","protected":false},"author":22,"featured_media":121159,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-121156","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-practice-question-videos","7":"page_type-video","8":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/121156","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\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=121156"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/121156\/revisions"}],"predecessor-version":[{"id":283555,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/121156\/revisions\/283555"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/121159"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=121156"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}