{"id":239749,"date":"2025-07-03T04:45:50","date_gmt":"2025-07-03T04:45:50","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=239749"},"modified":"2025-07-03T04:45:52","modified_gmt":"2025-07-03T04:45:52","slug":"what-is-the-quotient-in-simplest-form","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/03\/what-is-the-quotient-in-simplest-form\/","title":{"rendered":"What is the quotient in simplest form"},"content":{"rendered":"\n
What is the quotient in simplest form? State the restrictions on the variable. (z^2-4)\/(z-3) divided by (z+2)\/(z^2+z-12) I got z^2+2z-8, but I am not sure of the restrictions. Maybe 3 and -2.<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
Let’s break this down step by step.<\/p>\n\n\n\n
You are given the expression:z2\u22124z\u22123\u00f7z+2z2+z\u221212\\frac{{z^2 – 4}}{{z – 3}} \\div \\frac{{z + 2}}{{z^2 + z – 12}}z\u22123z2\u22124\u200b\u00f7z2+z\u221212z+2\u200b<\/p>\n\n\n\n
To divide fractions, we multiply by the reciprocal of the second fraction. So, this becomes:z2\u22124z\u22123\u00d7z2+z\u221212z+2\\frac{{z^2 – 4}}{{z – 3}} \\times \\frac{{z^2 + z – 12}}{{z + 2}}z\u22123z2\u22124\u200b\u00d7z+2z2+z\u221212\u200b<\/p>\n\n\n\n
Step 1: Factor the terms<\/h3>\n\n\n\n
\n
z2\u22124z^2 – 4z2\u22124 is a difference of squares, so it factors as:<\/li>\n<\/ul>\n\n\n\n
z2\u22124=(z\u22122)(z+2)z^2 – 4 = (z – 2)(z + 2)z2\u22124=(z\u22122)(z+2)<\/p>\n\n\n\n
\n
z2+z\u221212z^2 + z – 12z2+z\u221212 can be factored by looking for two numbers that multiply to -12 and add up to 1. These numbers are 4 and -3, so we have:<\/li>\n<\/ul>\n\n\n\n
z2+z\u221212=(z+4)(z\u22123)z^2 + z – 12 = (z + 4)(z – 3)z2+z\u221212=(z+4)(z\u22123)<\/p>\n\n\n\n
Now, substitute these factored forms into the expression:(z\u22122)(z+2)z\u22123\u00d7(z+4)(z\u22123)z+2\\frac{{(z – 2)(z + 2)}}{{z – 3}} \\times \\frac{{(z + 4)(z – 3)}}{{z + 2}}z\u22123(z\u22122)(z+2)\u200b\u00d7z+2(z+4)(z\u22123)\u200b<\/p>\n\n\n\n
Step 2: Cancel out common factors<\/h3>\n\n\n\n
Notice that z+2z + 2z+2 appears in both the numerator and denominator, and z\u22123z – 3z\u22123 also appears in both the numerator and denominator. So, we can cancel these terms:=(z\u22122)(z+4)1= \\frac{{(z – 2)(z + 4)}}{1}=1(z\u22122)(z+4)\u200b<\/p>\n\n\n\n
Which simplifies to:=(z\u22122)(z+4)= (z – 2)(z + 4)=(z\u22122)(z+4)<\/p>\n\n\n\n
So, your result is correct: the quotient is z2+2z\u22128z^2 + 2z – 8z2+2z\u22128<\/strong>.<\/p>\n\n\n\n
Step 3: Find the restrictions on the variable<\/h3>\n\n\n\n
To find the restrictions, we must look at the values of zzz that would make any denominator equal to zero. From the original expression, the denominators are z\u22123z – 3z\u22123 and z+2z + 2z+2, as well as the denominator z2+z\u221212z^2 + z – 12z2+z\u221212 that factored to (z\u22123)(z+4)(z – 3)(z + 4)(z\u22123)(z+4).<\/p>\n\n\n\n
Thus, the restrictions are z\u22603z \\neq 3z\ue020=3, z\u2260\u22122z \\neq -2z\ue020=\u22122, and z\u2260\u22124z \\neq -4z\ue020=\u22124<\/strong>, because these values would make the denominator zero and the expression undefined.<\/p>\n\n\n\n
What is the quotient in simplest form? State the restrictions on the variable. (z^2-4)\/(z-3) divided by (z+2)\/(z^2+z-12) I got z^2+2z-8, but I am not sure of the restrictions. Maybe 3 and -2. The Correct Answer and Explanation is: Let’s break this down step by step. You are given the expression:z2\u22124z\u22123\u00f7z+2z2+z\u221212\\frac{{z^2 – 4}}{{z – 3}} \\div […]<\/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-239749","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/239749","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=239749"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/239749\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=239749"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=239749"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=239749"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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