{"id":243316,"date":"2025-07-04T13:29:38","date_gmt":"2025-07-04T13:29:38","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=243316"},"modified":"2025-07-04T13:29:40","modified_gmt":"2025-07-04T13:29:40","slug":"evaluate-the-polynomial-y-x3-5x2-6x-0-55-at-x1-37","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/evaluate-the-polynomial-y-x3-5x2-6x-0-55-at-x1-37\/","title":{"rendered":"Evaluate the polynomial y= x^3- 5x^2+ 6x + 0.55 at x=1.37"},"content":{"rendered":"\n

Evaluate the polynomial y= x^3- 5x^2+ 6x + 0.55 at x=1.37 Use 3-digit arithmetic with rounding. Evaluate the percent relative error. (b) Repeat (a) but express y asy= ((x- 5)x +6)x + 0.55<\/p>\n\n\n\n

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

Let’s evaluate the polynomial y=x3\u22125×2+6x+0.55y = x^3 – 5x^2 + 6x + 0.55y=x3\u22125×2+6x+0.55 at x=1.37x = 1.37x=1.37 using 3-digit arithmetic and rounding. Then we’ll calculate the percent relative error. For part (b), we’ll repeat the process using the factored form y=((x\u22125)x+6)x+0.55y = ((x – 5)x + 6)x + 0.55y=((x\u22125)x+6)x+0.55.<\/p>\n\n\n\n

Part (a) – Polynomial Evaluation<\/h3>\n\n\n\n

We are given the polynomial:y=x3\u22125×2+6x+0.55y = x^3 – 5x^2 + 6x + 0.55y=x3\u22125×2+6x+0.55<\/p>\n\n\n\n

Substitute x=1.37x = 1.37x=1.37 into the polynomial:y=(1.37)3\u22125(1.37)2+6(1.37)+0.55y = (1.37)^3 – 5(1.37)^2 + 6(1.37) + 0.55y=(1.37)3\u22125(1.37)2+6(1.37)+0.55<\/p>\n\n\n\n

    \n
  1. (1.37)3=2.599073(1.37)^3 = 2.599073(1.37)3=2.599073<\/li>\n\n\n\n
  2. (1.37)2=1.8769(1.37)^2 = 1.8769(1.37)2=1.8769, so 5(1.37)2=9.38455(1.37)^2 = 9.38455(1.37)2=9.3845<\/li>\n\n\n\n
  3. 6(1.37)=8.226(1.37) = 8.226(1.37)=8.22<\/li>\n<\/ol>\n\n\n\n

    Now substitute these into the polynomial:y=2.599073\u22129.3845+8.22+0.55y = 2.599073 – 9.3845 + 8.22 + 0.55y=2.599073\u22129.3845+8.22+0.55y=2.599073+8.22\u22129.3845+0.55y = 2.599073 + 8.22 – 9.3845 + 0.55y=2.599073+8.22\u22129.3845+0.55y=10.819073\u22129.3845+0.55=2.984573y = 10.819073 – 9.3845 + 0.55 = 2.984573y=10.819073\u22129.3845+0.55=2.984573<\/p>\n\n\n\n

    Rounding to three significant digits:y\u22482.98y \\approx 2.98y\u22482.98<\/p>\n\n\n\n

    Percent Relative Error<\/h3>\n\n\n\n

    The exact value is calculated using a more precise value for the power terms:<\/p>\n\n\n\n

      \n
    1. (1.37)3=2.599073(1.37)^3 = 2.599073(1.37)3=2.599073<\/li>\n\n\n\n
    2. 5(1.37)2=9.38455(1.37)^2 = 9.38455(1.37)2=9.3845<\/li>\n\n\n\n
    3. 6(1.37)=8.226(1.37) = 8.226(1.37)=8.22<\/li>\n<\/ol>\n\n\n\n

      For the exact value, the polynomial sum is 2.599073\u22129.3845+8.22+0.55\u22482.9845732.599073 – 9.3845 + 8.22 + 0.55 \\approx 2.9845732.599073\u22129.3845+8.22+0.55\u22482.984573. The value rounded to three digits is 2.98.<\/p>\n\n\n\n

      The percent relative error is given by the formula:Percent Relative Error=\u2223Exact Value\u2212Approximate ValueExact Value\u2223\u00d7100\\text{Percent Relative Error} = \\left| \\frac{\\text{Exact Value} – \\text{Approximate Value}}{\\text{Exact Value}} \\right| \\times 100Percent Relative Error=\u200bExact ValueExact Value\u2212Approximate Value\u200b\u200b\u00d7100Percent Relative Error=\u22232.984573\u22122.982.984573\u2223\u00d7100\\text{Percent Relative Error} = \\left| \\frac{2.984573 – 2.98}{2.984573} \\right| \\times 100Percent Relative Error=\u200b2.9845732.984573\u22122.98\u200b\u200b\u00d7100Percent Relative Error\u2248\u22230.0045732.984573\u2223\u00d7100\u22480.153%\\text{Percent Relative Error} \\approx \\left| \\frac{0.004573}{2.984573} \\right| \\times 100 \\approx 0.153\\%Percent Relative Error\u2248\u200b2.9845730.004573\u200b\u200b\u00d7100\u22480.153%<\/p>\n\n\n\n

      Part (b) – Using the Factored Form<\/h3>\n\n\n\n

      Now, we express the polynomial in factored form:y=((x\u22125)x+6)x+0.55y = ((x – 5)x + 6)x + 0.55y=((x\u22125)x+6)x+0.55<\/p>\n\n\n\n

      Substitute x=1.37x = 1.37x=1.37 into this form:y=((1.37\u22125)\u00d71.37+6)\u00d71.37+0.55y = ((1.37 – 5) \\times 1.37 + 6) \\times 1.37 + 0.55y=((1.37\u22125)\u00d71.37+6)\u00d71.37+0.55<\/p>\n\n\n\n

      First, calculate the inner expressions:1.37\u22125=\u22123.631.37 – 5 = -3.631.37\u22125=\u22123.63(\u22123.63)\u00d71.37=\u22124.9781(-3.63) \\times 1.37 = -4.9781(\u22123.63)\u00d71.37=\u22124.9781\u22124.9781+6=1.0219-4.9781 + 6 = 1.0219\u22124.9781+6=1.02191.0219\u00d71.37=1.3988031.0219 \\times 1.37 = 1.3988031.0219\u00d71.37=1.3988031.398803+0.55=1.9488031.398803 + 0.55 = 1.9488031.398803+0.55=1.948803<\/p>\n\n\n\n

      Rounding to three significant digits:y\u22481.95y \\approx 1.95y\u22481.95<\/p>\n\n\n\n

      Percent Relative Error for Factored Form<\/h3>\n\n\n\n

      Now calculate the percent relative error:Percent Relative Error=\u2223Exact Value\u2212Approximate ValueExact Value\u2223\u00d7100\\text{Percent Relative Error} = \\left| \\frac{\\text{Exact Value} – \\text{Approximate Value}}{\\text{Exact Value}} \\right| \\times 100Percent Relative Error=\u200bExact ValueExact Value\u2212Approximate Value\u200b\u200b\u00d7100<\/p>\n\n\n\n

      The exact value is 2.984573, and the approximate value is 1.95:Percent Relative Error=\u22232.984573\u22121.952.984573\u2223\u00d7100\\text{Percent Relative Error} = \\left| \\frac{2.984573 – 1.95}{2.984573} \\right| \\times 100Percent Relative Error=\u200b2.9845732.984573\u22121.95\u200b\u200b\u00d7100Percent Relative Error=\u22231.0345732.984573\u2223\u00d7100\u224834.7%\\text{Percent Relative Error} = \\left| \\frac{1.034573}{2.984573} \\right| \\times 100 \\approx 34.7\\%Percent Relative Error=\u200b2.9845731.034573\u200b\u200b\u00d7100\u224834.7%<\/p>\n\n\n\n

      Conclusion<\/h3>\n\n\n\n
        \n
      • In part (a), the evaluation of the polynomial gives y\u22482.98y \\approx 2.98y\u22482.98, with a percent relative error of about 0.153%0.153\\%0.153%.<\/li>\n\n\n\n
      • In part (b), using the factored form gives y\u22481.95y \\approx 1.95y\u22481.95, with a percent relative error of about 34.7%34.7\\%34.7%.<\/li>\n<\/ul>\n\n\n\n

        This significant difference in errors between the two forms highlights how the polynomial’s structure can affect the accuracy of numerical evaluations. The original polynomial gives a much more accurate result than the factored form in this case.<\/p>\n\n\n\n

        \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

        Evaluate the polynomial y= x^3- 5x^2+ 6x + 0.55 at x=1.37 Use 3-digit arithmetic with rounding. Evaluate the percent relative error. (b) Repeat (a) but express y asy= ((x- 5)x +6)x + 0.55 The Correct Answer and Explanation is: Let’s evaluate the polynomial y=x3\u22125×2+6x+0.55y = x^3 – 5x^2 + 6x + 0.55y=x3\u22125×2+6x+0.55 at x=1.37x = […]<\/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-243316","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243316","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=243316"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243316\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=243316"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=243316"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=243316"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}