{"id":47675,"date":"2025-07-02T14:54:10","date_gmt":"2025-07-02T14:54:10","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=47675"},"modified":"2025-07-02T14:54:11","modified_gmt":"2025-07-02T14:54:11","slug":"suppose-that-fx-and-gx-are-given-by-the-power-series","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/suppose-that-fx-and-gx-are-given-by-the-power-series\/","title":{"rendered":"Suppose that f(x) and g(x) are given by the power series"},"content":{"rendered":"\n<p>Suppose that f(x) and g(x) are given by the power series f(x) = 2 + 3x + 3x^2 + 2x^3 and g(x) = 8 + 16x + 24x^2 + 31x^3 + B. By dividing the power series, find the first few terms of the series for the quotient g(x)\/h(x) = Co + C1x + C2x^2 + C3x^3 + \u2026 Co = C1 = 2 C2 = 3 C3 = \u2026<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1006\" height=\"896\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/image-130.png\" alt=\"\" class=\"wp-image-47676\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/image-130.png 1006w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/image-130-300x267.png 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/image-130-768x684.png 768w\" sizes=\"auto, (max-width: 1006px) 100vw, 1006px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Based on the provided power series, the correct coefficients for h(x) are:<br>c\u2080 = 4<br>c\u2081 = 2<br>c\u2082 = 3<br>c\u2083 = 4<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>To find the coefficients of the power series for the quotient h(x) = g(x) \/ f(x), we can use the equivalent relationship g(x) = f(x) * h(x). This allows us to determine the unknown coefficients (c\u2080, c\u2081, c\u2082, c\u2083, &#8230;) by expanding the product of f(x) and h(x) and then equating the coefficients of corresponding powers of x with those in g(x).<\/p>\n\n\n\n<p>The given power series are:<br>f(x) = 2 + 3x + 3x\u00b2 + 2x\u00b3 + &#8230;<br>g(x) = 8 + 16x + 24x\u00b2 + 31x\u00b3 + &#8230;<br>And the quotient series is:<br>h(x) = c\u2080 + c\u2081x + c\u2082x\u00b2 + c\u2083x\u00b3 + &#8230;<\/p>\n\n\n\n<p>Setting up the equation g(x) = f(x)h(x):<br>8 + 16x + 24x\u00b2 + 31x\u00b3 + &#8230; = (2 + 3x + 3x\u00b2 + 2x\u00b3 + &#8230;)(c\u2080 + c\u2081x + c\u2082x\u00b2 + c\u2083x\u00b3 + &#8230;)<\/p>\n\n\n\n<p>Now, we compare the coefficients for each power of x, solving for one coefficient at a time.<\/p>\n\n\n\n<p><strong>Constant term (x\u2070):<\/strong><br>The constant term on the left is 8. On the right, it&#8217;s the product of the constant terms, 2 * c\u2080.<br>8 = 2c\u2080<br><strong>c\u2080 = 4<\/strong><\/p>\n\n\n\n<p><strong>Coefficient of x (x\u00b9):<\/strong><br>The coefficient on the left is 16. On the right, the x term is formed by (2 * c\u2081x) and (3x * c\u2080).<br>16 = 2c\u2081 + 3c\u2080<br>Substitute the value of c\u2080 = 4:<br>16 = 2c\u2081 + 3(4)<br>16 = 2c\u2081 + 12<br>4 = 2c\u2081<br><strong>c\u2081 = 2<\/strong><\/p>\n\n\n\n<p><strong>Coefficient of x\u00b2:<\/strong><br>The coefficient on the left is 24. On the right, the x\u00b2 term is formed by (2 * c\u2082x\u00b2), (3x * c\u2081x), and (3x\u00b2 * c\u2080).<br>24 = 2c\u2082 + 3c\u2081 + 3c\u2080<br>Substitute the values of c\u2080 = 4 and c\u2081 = 2:<br>24 = 2c\u2082 + 3(2) + 3(4)<br>24 = 2c\u2082 + 6 + 12<br>24 = 2c\u2082 + 18<br>6 = 2c\u2082<br><strong>c\u2082 = 3<\/strong><\/p>\n\n\n\n<p><strong>Coefficient of x\u00b3:<\/strong><br>The coefficient on the left is 31. On the right, the x\u00b3 term is formed by (2 * c\u2083x\u00b3), (3x * c\u2082x\u00b2), (3x\u00b2 * c\u2081x), and (2x\u00b3 * c\u2080).<br>31 = 2c\u2083 + 3c\u2082 + 3c\u2081 + 2c\u2080<br>Substitute the values of c\u2080 = 4, c\u2081 = 2, and c\u2082 = 3:<br>31 = 2c\u2083 + 3(3) + 3(2) + 2(4)<br>31 = 2c\u2083 + 9 + 6 + 8<br>31 = 2c\u2083 + 23<br>8 = 2c\u2083<br><strong>c\u2083 = 4<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-265.jpeg\" alt=\"\" class=\"wp-image-47677\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-265.jpeg 1024w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-265-300x300.jpeg 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-265-150x150.jpeg 150w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-265-768x768.jpeg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Suppose that f(x) and g(x) are given by the power series f(x) = 2 + 3x + 3x^2 + 2x^3 and g(x) = 8 + 16x + 24x^2 + 31x^3 + B. By dividing the power series, find the first few terms of the series for the quotient g(x)\/h(x) = Co + C1x + C2x^2 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-47675","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/47675","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=47675"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/47675\/revisions"}],"predecessor-version":[{"id":47678,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/47675\/revisions\/47678"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=47675"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=47675"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=47675"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}