{"id":146559,"date":"2024-10-01T07:56:36","date_gmt":"2024-10-01T07:56:36","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=146559"},"modified":"2024-10-01T07:56:39","modified_gmt":"2024-10-01T07:56:39","slug":"does-the-expression-x3-1-x2-1-simplify-to-x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/01\/does-the-expression-x3-1-x2-1-simplify-to-x\/","title":{"rendered":"Does the expression x^3-1\/x^2-1 simplify to x"},"content":{"rendered":"\n

Does the expression x^3-1\/x^2-1 simplify to x?
No, because x3 \u2013 1 can be factored as x(x2 \u2013 x + 1) and x2 \u2013 1 can be factored as x(x \u2013 1), so only x can be canceled.
Yes, because x3(x \u2013 1) can be factored as x2(x \u2013 1) and x2 \u2013 1 can be factored as x(x \u2013 1), so (x \u2013 1) can be canceled.
No, because the \u20131 in the numerator and denominator is not a common factor and cannot be canceled.
Yes, because \u20131 in the numerator and denominator is a common factor and can be canceled.<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

The expression (\\frac{x^3 – 1}{x^2 – 1}) does not simplify to (x). Let’s analyze this step by step.<\/p>\n\n\n\n

    \n
  1. Factoring the Expressions:<\/strong><\/li>\n<\/ol>\n\n\n\n
      \n
    • The numerator (x^3 – 1) can be factored using the difference of cubes formula, which states (a^3 – b^3 = (a – b)(a^2 + ab + b^2)). Here, (a = x) and (b = 1):
      [
      x^3 – 1 = (x – 1)(x^2 + x + 1)
      ]<\/li>\n\n\n\n
    • The denominator (x^2 – 1) can be factored using the difference of squares formula, which states (a^2 – b^2 = (a – b)(a + b)). Here, (a = x) and (b = 1):
      [
      x^2 – 1 = (x – 1)(x + 1)
      ]<\/li>\n<\/ul>\n\n\n\n
        \n
      1. Rewriting the Expression:<\/strong>
        Substituting the factored forms back into the original expression gives:
        [
        \\frac{x^3 – 1}{x^2 – 1} = \\frac{(x – 1)(x^2 + x + 1)}{(x – 1)(x + 1)}
        ]<\/li>\n\n\n\n
      2. Canceling Common Factors:<\/strong>
        We can cancel the common factor (x – 1) from the numerator and denominator, assuming (x \\neq 1):
        [
        \\frac{x^2 + x + 1}{x + 1}
        ]<\/li>\n\n\n\n
      3. Final Simplification:<\/strong>
        After canceling, the expression simplifies to:
        [
        \\frac{x^2 + x + 1}{x + 1}
        ]
        This does not simplify to (x). Instead, we can analyze this expression further, but it won’t yield (x) directly.<\/li>\n\n\n\n
      4. Conclusion:<\/strong>
        Therefore, the statement that (\\frac{x^3 – 1}{x^2 – 1} = x) is incorrect. The common factor (x – 1) can be canceled, but it does not lead to the conclusion that the entire expression simplifies to (x). Instead, it simplifies to (\\frac{x^2 + x + 1}{x + 1}), which is a distinct rational expression.<\/li>\n<\/ol>\n\n\n\n

        In summary, the correct answer is No<\/strong>, the expression does not simplify to (x), as it simplifies to (\\frac{x^2 + x + 1}{x + 1}).<\/p>\n","protected":false},"excerpt":{"rendered":"

        Does the expression x^3-1\/x^2-1 simplify to x?No, because x3 \u2013 1 can be factored as x(x2 \u2013 x + 1) and x2 \u2013 1 can be factored as x(x \u2013 1), so only x can be canceled.Yes, because x3(x \u2013 1) can be factored as x2(x \u2013 1) and x2 \u2013 1 can be factored […]<\/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-146559","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/146559","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=146559"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/146559\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=146559"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=146559"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=146559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}