{"id":181569,"date":"2025-01-11T02:42:31","date_gmt":"2025-01-11T02:42:31","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=181569"},"modified":"2025-01-11T02:42:33","modified_gmt":"2025-01-11T02:42:33","slug":"1-number-4-is-a-solution-to-the-equation-because-the-remainder-of-the-division","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/11\/1-number-4-is-a-solution-to-the-equation-because-the-remainder-of-the-division\/","title":{"rendered":"1.number -4 is a solution to the equation because the remainder of the division"},"content":{"rendered":"\n

1.number -4 is a solution to the equation because the remainder of the division, x^(3)+18x^(2)+101x+180 divided by x+4<\/p>\n\n\n\n

2.iris has 120 songs on her playlist. each week, she downloads 4 new song<\/p>\n\n\n\n

3.X ~ N(0,1); P(X > .25 or X <><\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Let’s go through each statement and explain them.<\/p>\n\n\n\n

1. Remainder of Polynomial Division:<\/strong><\/h3>\n\n\n\n

The first statement suggests that \u22124-4 is a solution to the equation because the remainder of dividing the polynomial x3+18×2+101x+180x^3 + 18x^2 + 101x + 180 by x+4x + 4 is zero.<\/p>\n\n\n\n

    \n
  • Explanation:<\/strong> When you divide a polynomial by a binomial of the form x\u2212ax – a, the remainder theorem states that the remainder is equal to the value of the polynomial evaluated at x=ax = a. In this case, you are dividing by x+4x + 4, which is the same as x\u2212(\u22124)x – (-4). Therefore, to check if \u22124-4 is a root of the polynomial, substitute x=\u22124x = -4 into the polynomial x3+18×2+101x+180x^3 + 18x^2 + 101x + 180. Let’s substitute: P(\u22124)=(\u22124)3+18(\u22124)2+101(\u22124)+180P(-4) = (-4)^3 + 18(-4)^2 + 101(-4) + 180 =\u221264+18(16)\u2212404+180= -64 + 18(16) – 404 + 180 =\u221264+288\u2212404+180= -64 + 288 – 404 + 180 =0= 0 Since P(\u22124)=0P(-4) = 0, the remainder is zero, confirming that \u22124-4 is indeed a solution to the equation.<\/li>\n<\/ul>\n\n\n\n

    2. Iris\u2019s Playlist:<\/strong><\/h3>\n\n\n\n

    The second statement says Iris has 120 songs on her playlist, and each week she downloads 4 new songs.<\/p>\n\n\n\n

      \n
    • Explanation:<\/strong> To find out how many songs Iris will have after a certain number of weeks, we can use a simple linear model. Let the number of weeks be nn, and the total number of songs after nn weeks be S(n)S(n). The total number of songs increases by 4 songs each week, starting from 120 songs. The formula for the total number of songs after nn weeks is: S(n)=120+4nS(n) = 120 + 4n If we wanted to know how many songs she has after 5 weeks, for example, we would substitute n=5n = 5: S(5)=120+4(5)=120+20=140S(5) = 120 + 4(5) = 120 + 20 = 140 So, after 5 weeks, Iris will have 140 songs.<\/li>\n<\/ul>\n\n\n\n

      3. Probability Statement:<\/strong><\/h3>\n\n\n\n

      The third statement involves a probability question: X\u223cN(0,1)X \\sim N(0,1), and we are asked to find P(X>0.25 or X<\u22120.25)P(X > 0.25 \\text{ or } X < -0.25).<\/p>\n\n\n\n

        \n
      • Explanation:<\/strong> The symbol X\u223cN(0,1)X \\sim N(0, 1) means that XX follows a standard normal distribution, where the mean is 0 and the standard deviation is 1. The probability expression asks for the likelihood that XX is either greater than 0.25 or less than -0.25. Using symmetry of the standard normal distribution:
        • P(X>0.25)P(X > 0.25) is the probability that XX is greater than 0.25.<\/li>
        • P(X<\u22120.25)P(X < -0.25) is the probability that XX is less than -0.25.<\/li><\/ul>The probability for these two events can be found by looking up the corresponding z-scores in a standard normal distribution table, or using a calculator: P(X>0.25)=1\u2212P(X\u22640.25)\u22481\u22120.5987=0.4013P(X > 0.25) = 1 – P(X \\leq 0.25) \\approx 1 – 0.5987 = 0.4013 P(X<\u22120.25)=P(X\u2264\u22120.25)\u22480.4013P(X < -0.25) = P(X \\leq -0.25) \\approx 0.4013 Since the two events are mutually exclusive (i.e., they cannot both occur at the same time), the total probability is the sum of these two probabilities: P(X>0.25\u00a0or\u00a0X<\u22120.25)=0.4013+0.4013=0.8026P(X > 0.25 \\text{ or } X < -0.25) = 0.4013 + 0.4013 = 0.8026 Therefore, the probability is approximately 0.8026.<\/li>\n<\/ul>\n\n\n\n

          In summary:<\/p>\n\n\n\n

            \n
          • The remainder theorem confirms that \u22124-4 is a solution.<\/li>\n\n\n\n
          • Iris\u2019s playlist grows linearly by 4 songs per week.<\/li>\n\n\n\n
          • The probability of XX being greater than 0.25 or less than -0.25 is approximately 0.8026.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

            1.number -4 is a solution to the equation because the remainder of the division, x^(3)+18x^(2)+101x+180 divided by x+4 2.iris has 120 songs on her playlist. each week, she downloads 4 new song 3.X ~ N(0,1); P(X > .25 or X <> The correct answer and explanation is: Let’s go through each statement and explain them. […]<\/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-181569","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/181569","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=181569"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/181569\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=181569"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=181569"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=181569"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}