{"id":216716,"date":"2025-05-20T09:05:48","date_gmt":"2025-05-20T09:05:48","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=216716"},"modified":"2025-05-20T09:05:50","modified_gmt":"2025-05-20T09:05:50","slug":"write-a-polynomial-that-represents-the-length-of-the-rectangle","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/20\/write-a-polynomial-that-represents-the-length-of-the-rectangle\/","title":{"rendered":"Write a polynomial that represents the length of the rectangle."},"content":{"rendered":"\n<p>Write a polynomial that represents the length of the rectangle.<\/p>\n\n\n\n<p>The width is x+0.2 ?units.<\/p>\n\n\n\n<p>The area is 0.8&#215;3+0.56&#215;2+0.58x+0.1 ?square units.<\/p>\n\n\n\n<p>The length is units.<\/p>\n\n\n\n<p>(Use integers or decimals for any numbers in the expression.)<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the <strong>length<\/strong> of the rectangle, we use the relationship between <strong>area<\/strong>, <strong>length<\/strong>, and <strong>width<\/strong> of a rectangle: Area=Length\u00d7Width\\text{Area} = \\text{Length} \\times \\text{Width}<\/p>\n\n\n\n<p>We are given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Width<\/strong> = x+0.2x + 0.2<\/li>\n\n\n\n<li><strong>Area<\/strong> = 0.8&#215;3+0.56&#215;2+0.58x+0.10.8x^3 + 0.56x^2 + 0.58x + 0.1<\/li>\n<\/ul>\n\n\n\n<p>We need to <strong>solve for length<\/strong>, so we rearrange the formula: Length=AreaWidth=0.8&#215;3+0.56&#215;2+0.58x+0.1x+0.2\\text{Length} = \\frac{\\text{Area}}{\\text{Width}} = \\frac{0.8x^3 + 0.56x^2 + 0.58x + 0.1}{x + 0.2}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step-by-step Polynomial Division:<\/h3>\n\n\n\n<p>We&#8217;ll divide the cubic polynomial by the linear binomial using <strong>polynomial long division<\/strong>:<\/p>\n\n\n\n<p><strong>Divide:<\/strong><br>0.8&#215;3\u00f7x=0.8&#215;20.8x^3 \\div x = 0.8x^2<\/p>\n\n\n\n<p><strong>Multiply and Subtract:<\/strong> (x+0.2)(0.8&#215;2)=0.8&#215;3+0.16&#215;2(x + 0.2)(0.8x^2) = 0.8x^3 + 0.16x^2<\/p>\n\n\n\n<p>Subtract: (0.8&#215;3+0.56&#215;2)\u2212(0.8&#215;3+0.16&#215;2)=0.4&#215;2(0.8x^3 + 0.56x^2) &#8211; (0.8x^3 + 0.16x^2) = 0.4x^2<\/p>\n\n\n\n<p><strong>Bring down:<\/strong> +0.58x<\/p>\n\n\n\n<p><strong>Next term:<\/strong><br>0.4&#215;2\u00f7x=0.4&#215;0.4x^2 \\div x = 0.4x<\/p>\n\n\n\n<p><strong>Multiply:<\/strong><br>(x+0.2)(0.4x)=0.4&#215;2+0.08x(x + 0.2)(0.4x) = 0.4x^2 + 0.08x<\/p>\n\n\n\n<p>Subtract: (0.4&#215;2+0.58x)\u2212(0.4&#215;2+0.08x)=0.5x(0.4x^2 + 0.58x) &#8211; (0.4x^2 + 0.08x) = 0.5x<\/p>\n\n\n\n<p><strong>Bring down:<\/strong> +0.1<\/p>\n\n\n\n<p><strong>Next term:<\/strong><br>0.5x\u00f7x=0.50.5x \\div x = 0.5<\/p>\n\n\n\n<p><strong>Multiply:<\/strong><br>(x+0.2)(0.5)=0.5x+0.1(x + 0.2)(0.5) = 0.5x + 0.1<\/p>\n\n\n\n<p>Subtract: (0.5x+0.1)\u2212(0.5x+0.1)=0(0.5x + 0.1) &#8211; (0.5x + 0.1) = 0<\/p>\n\n\n\n<p>So, there is <strong>no remainder<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>Length=0.8&#215;2+0.4x+0.5\\text{Length} = 0.8x^2 + 0.4x + 0.5<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation (300+ words):<\/h3>\n\n\n\n<p>To determine the length of the rectangle when the area and width are known, we use a fundamental geometric formula:<br><strong>Area = Length \u00d7 Width<\/strong>.<\/p>\n\n\n\n<p>Rearranging the formula, we isolate <strong>Length<\/strong>: Length=AreaWidth\\text{Length} = \\frac{\\text{Area}}{\\text{Width}}<\/p>\n\n\n\n<p>Here, the area is given as a cubic polynomial: 0.8&#215;3+0.56&#215;2+0.58x+0.10.8x^3 + 0.56x^2 + 0.58x + 0.1<\/p>\n\n\n\n<p>And the width is a binomial expression: x+0.2x + 0.2<\/p>\n\n\n\n<p>To find the quotient (which represents the length), we divide the area polynomial by the width polynomial. This is done using polynomial division. In this method, we divide the highest-degree term in the numerator by the highest-degree term in the denominator to determine the first term of the quotient. Then we multiply and subtract successively until we reduce the polynomial or reach a remainder of zero.<\/p>\n\n\n\n<p>In this case, the polynomial division yields: 0.8&#215;2+0.4x+0.50.8x^2 + 0.4x + 0.5<\/p>\n\n\n\n<p>This expression is the <strong>length of the rectangle<\/strong>, as multiplying it by the width x+0.2x + 0.2 would regenerate the original area polynomial. Also, since the division resulted in no remainder, this confirms that the width is a true factor of the area, and the result is a valid expression for length.<\/p>\n\n\n\n<p>This process of dividing one polynomial by another is crucial in algebra for simplifying expressions and understanding relationships among geometric dimensions expressed algebraically.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Write a polynomial that represents the length of the rectangle. The width is x+0.2 ?units. The area is 0.8&#215;3+0.56&#215;2+0.58x+0.1 ?square units. The length is units. (Use integers or decimals for any numbers in the expression.) The Correct Answer and Explanation is: To find the length of the rectangle, we use the relationship between area, length, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-216716","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216716","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=216716"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216716\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=216716"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=216716"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=216716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}