{"id":203956,"date":"2025-03-21T06:53:35","date_gmt":"2025-03-21T06:53:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=203956"},"modified":"2025-03-21T06:53:37","modified_gmt":"2025-03-21T06:53:37","slug":"factor-and-simplify-to-obtain-a-product-of-sums-expression-with-four-term","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/21\/factor-and-simplify-to-obtain-a-product-of-sums-expression-with-four-term\/","title":{"rendered":"Factor and simplify to obtain a product-of-sums expression with four term"},"content":{"rendered":"\n<p>Factor and simplify to obtain a product-of-sums expression with four terms: abc\u2032 + d\u2032e + ace + b\u2032c\u2032d\u2032<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To simplify the given Boolean expression and express it in a product-of-sums (POS) form with four terms, we follow these steps:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Given Boolean Expression:<\/h3>\n\n\n\n<p>[<br>abc&#8217; + d&#8217;e + ace + b&#8217;c&#8217;d&#8217;<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Identify Common Factors<\/h3>\n\n\n\n<p>We analyze the terms to group and factor common variables:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Grouping terms strategically:<br>[<br>(abc&#8217; + ace) + (d&#8217;e + b&#8217;c&#8217;d&#8217;)<br>]<\/li>\n\n\n\n<li>Factor within each group:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>From <strong>(abc&#8217; + ace)<\/strong>, factor out <strong>(a)<\/strong>:<br>[<br>a(bc&#8217; + ce)<br>]<\/li>\n\n\n\n<li>From <strong>(d&#8217;e + b&#8217;c&#8217;d&#8217;)<\/strong>, factor out <strong>(d&#8217;)<\/strong>:<br>[<br>d'(e + b&#8217;c&#8217;)<br>]<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Further Factorization<\/h3>\n\n\n\n<p>Looking at the term ( bc&#8217; + ce ), we try factoring in a way that leads to a sum of products:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Rewriting<\/strong> ( bc&#8217; + ce ):<br>[<br>bc&#8217; + ce = c'(b) + c(e) = c'(b) + c(e)<br>]<\/li>\n\n\n\n<li><strong>Rewriting<\/strong> ( e + b&#8217;c&#8217; ) (which is already in sum form).<\/li>\n<\/ol>\n\n\n\n<p>Thus, our new factored expression is:<\/p>\n\n\n\n<p>[<br>(a(c&#8217; + e)(b + c)) (d&#8217; + (e + b&#8217;c&#8217;))<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Expression in Product-of-Sums Form:<\/h3>\n\n\n\n<p>[<br>(a + d&#8217;)(c&#8217; + e)(b + c)(e + b&#8217;c&#8217;)<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Factoring is crucial<\/strong>: We carefully group terms to extract common factors, simplifying step by step.<\/li>\n\n\n\n<li><strong>Converting to POS form<\/strong>: The goal is to express the equation as a product of sum terms, ensuring it follows the desired format.<\/li>\n\n\n\n<li><strong>Logical equivalence<\/strong>: The transformation preserves the original logic but presents it in a structured, minimal format.<\/li>\n<\/ol>\n\n\n\n<p>Thus, the correct <strong>Product-of-Sums<\/strong> expression is:<\/p>\n\n\n\n<p>[<br>(a + d&#8217;)(c&#8217; + e)(b + c)(e + b&#8217;c&#8217;)<br>]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Factor and simplify to obtain a product-of-sums expression with four terms: abc\u2032 + d\u2032e + ace + b\u2032c\u2032d\u2032 The correct answer and explanation is : To simplify the given Boolean expression and express it in a product-of-sums (POS) form with four terms, we follow these steps: Given Boolean Expression: [abc&#8217; + d&#8217;e + ace + [&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-203956","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203956","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=203956"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203956\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=203956"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=203956"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=203956"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}