{"id":664789,"date":"2023-07-15T14:16:47","date_gmt":"2023-07-15T08:46:47","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=664789"},"modified":"2024-03-06T14:29:53","modified_gmt":"2024-03-06T08:59:53","slug":"polynomials-2","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/","title":{"rendered":"Polynomials"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Introduction_to_Polynomials\" title=\"Introduction to Polynomials\">Introduction to Polynomials<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Expression_of_the_standard_polynomial\" title=\"Expression of the standard polynomial\">Expression of the standard polynomial<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Definition_of_Polynomials\" title=\"Definition of Polynomials\">Definition of Polynomials<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#What_is_the_meaning_of_Polynomial\" title=\"What is the meaning of Polynomial?\">What is the meaning of Polynomial?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Notation_of_Polynomials\" title=\"Notation of Polynomials\">Notation of Polynomials<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Properties_of_the_Polynomials\" title=\"Properties of the Polynomials\">Properties of the Polynomials<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Division_Algorithm\" title=\"Division Algorithm\">Division Algorithm<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Bezouts_Theorem\" title=\"Bezout\u2019s Theorem\">Bezout\u2019s Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Remainder_Theorem\" title=\"Remainder Theorem\">Remainder Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Factor_Theorem\" title=\"Factor Theorem\">Factor Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Intermediate_Value_Theorem\" title=\"Intermediate Value Theorem\">Intermediate Value Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Fundamental_Theorems_of_Algebra\" title=\"Fundamental Theorems of Algebra\">Fundamental Theorems of Algebra<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Frequently_Asked_Questions_on_Polynomials\" title=\"Frequently Asked Questions on Polynomials\">Frequently Asked Questions on Polynomials<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#What_is_a_Polynomial\" title=\"What is a Polynomial?\">What is a Polynomial?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#What_is_the_standard_form_of_the_polynomial\" title=\"What is the standard form of the polynomial?\">What is the standard form of the polynomial?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/polynomials\/#Is_6_a_polynomial\" title=\"Is 6 a polynomial?\">Is 6 a polynomial?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Introduction_to_Polynomials\"><\/span>Introduction to Polynomials<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Polynomials are defined as the type of algebraic expressions whose variables have only non-negative integers as their powers.<br \/>\nFor example:<br \/>\nf(x) = 5x\u00b2 &#8211; x + 1 is a polynomial.<\/p>\n<p>The algebraic expression 3x\u00b3 + 4x + 5\/x + 6x\u00b3\/\u00b2 is not a polynomial since one of the powers of &#8216;x&#8217; is a fraction and the other is negative.<\/p>\n<ul>\n<li>Algebraic expressions with the power in either negative or fraction aren&#8217;t considered Polynomials.<\/li>\n<li>The terms in the polynomial expression comprise the variables, exponents, and constants.<\/li>\n<li>The first term of the polynomial expression in the standard form is called the &#8220;leading term.&#8221;<\/li>\n<\/ul>\n<p>A <strong>standard polynomial<\/strong> is a polynomial whose first term is the highest power holder, and the subsequent terms are arranged in descending order of the powers.<br \/>\nThe number multiplied by a variable is called the &#8220;<strong>coefficient<\/strong>.&#8221; The number without any variable is called a &#8220;<strong>constant<\/strong>.&#8221;<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Expression_of_the_standard_polynomial\"><\/span>Expression of the standard polynomial<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>a<sup>xm<\/sup> + bx<sup>(m-1)<\/sup> + bx<sup>(m-2)<\/sup> + \u2026\u2026\u2026 + nx\u2070 is the standard polynomial.<br \/>\nWhere x\u2070 equals 1, therefore, the coefficient is a constant.<\/p>\n<p>In this article, we will discuss the properties of the polynomial functions.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Definition_of_Polynomials\"><\/span>Definition of Polynomials<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Polynomials are algebraic expressions that consist of a combination of variables and coefficients. Variables are also referred to as indeterminates.<br \/>\nWe can perform a number of arithmetic operations on polynomials, such as addition, subtraction, multiplication, and more.<\/p>\n<p>An example of a polynomial with one variable is x\u00b2+x-1. In this example, there are three terms: x\u00b2, x, and -1.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"What_is_the_meaning_of_Polynomial\"><\/span>What is the meaning of Polynomial?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The word polynomial is derived from the Greek words \u2018poly,\u2019 which means &#8216;many,&#8217; and \u2018nominal,\u2019 which means &#8216;terms,&#8217; so altogether, it means \u201cmany terms.\u201d A polynomial can have any number of terms but can never have an infinite number of terms.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Notation_of_Polynomials\"><\/span>Notation of Polynomials<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The polynomial function is denoted by P(x), where x represents the variable.<br \/>\nFor example, P(x) = x\u00b2-5x+9<\/p>\n<p>If the variable is represented by a, then the function will be P(a).<\/p>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/articles\/math-articles\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Math Articles<br \/>\n<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/formulas\/math-formulas\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Math Formulas<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/topics\/monomials-binomials-trinomials-and-polynomials\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Monomial, Binomials, Trinomials, and Polynomials<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/topics\/types-of-polynomials\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Polynomials and Its Types<\/button><\/a><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Properties_of_the_Polynomials\"><\/span>Properties of the Polynomials<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Some of the important properties of polynomials are mentioned below.<\/p>\n<p><strong>Property 1:<\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Division_Algorithm\"><\/span>Division Algorithm<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If a polynomial f(x) is divided by a polynomial g(x), then as result it gives us quotient Q(x) with remainder R(x), then,<\/p>\n<p>f(x) = g(x) \u2022 Q(x) + R(x)<\/p>\n<p>Where,<br \/>\nR(x)=0 or the degree of R(x) is lesser than the degree of g(x)<\/p>\n<p><strong>Property 2:<\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Bezouts_Theorem\"><\/span>Bezout\u2019s Theorem<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Bezout&#8217;s theorem states that a polynomial f(x) is divisible by a binomial (x \u2013 a) iff f(a) = 0.<\/p>\n<p><strong>Property 3:<\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Remainder_Theorem\"><\/span>Remainder Theorem<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The Remainder theorem states that if P(x) is divided by (x \u2013 a) with remainder r, then P(a) = r.<\/p>\n<p><strong>Property 4:<\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Factor_Theorem\"><\/span>Factor Theorem<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A polynomial f(x) divided by q(x) results in r(x) with zero remainder iff q(x) is a factor of f(x).<\/p>\n<p><strong>Property 5:<\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Intermediate_Value_Theorem\"><\/span>Intermediate Value Theorem<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If f(x) is a polynomial, and f(x) \u2260 f(y) for (x &lt; y), then f(x) takes every value from f(x) to f(y) in the closed interval [x, y].<\/p>\n<p><strong>Property 6<\/strong><\/p>\n<p>The addition, subtraction, and multiplication arithmetic operation on the polynomials P and Q, we get the resulting polynomial whose,<\/p>\n<p>Degree(P \u00b1 Q) \u2264 Degree(P or Q)<br \/>\nDegree(P \u00d7 Q) = Degree(P) + Degree(Q)<\/p>\n<p><strong>Property 7<\/strong><\/p>\n<p>If a polynomial f(x) is divisible by a polynomial g(x), then every zero of g(x) is also a zero of f(x).<\/p>\n<p><strong>Property 8<\/strong><\/p>\n<p>If a polynomial f(x) is divisible by two coprime polynomials g(x) and r(x), then it is divisible by (g(x) \u2022 r(x)).<\/p>\n<p><strong>Property 9<\/strong><\/p>\n<p>If f(x) = a0 + a1x + a0x\u00b2 + \u2026\u2026 + anxn is a polynomial such that deg(f(x)) = n \u2265 0, then f has at most \u201cn\u201d distinct roots.<\/p>\n<p><strong>Property 10: <\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Fundamental_Theorems_of_Algebra\"><\/span>Fundamental Theorems of Algebra<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Every non-constant single-variable polynomial with complex coefficients always has at least one complex zero.<\/p>\n<p><strong>Property 11<\/strong><\/p>\n<p>If f(x) is a polynomial with real coefficients and has one complex zero, say x = a \u2013 bi, then x = a + bi will also be a zero of f(x).<\/p>\n<p>Also check the <strong><a href=\"https:\/\/infinitylearn.com\/surge\/topics\/types-of-polynomials\/\">Types of polynomials<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Frequently_Asked_Questions_on_Polynomials\"><\/span>Frequently Asked Questions on Polynomials<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_a_Polynomial\"><\/span>What is a Polynomial?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tA polynomial is an expression consisting of variables (or indeterminate), exponents, and constants. For example, 3x2 -2x-10 is a polynomial. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_standard_form_of_the_polynomial\"><\/span>What is the standard form of the polynomial?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\t A standard polynomial is one where the highest degree is the first term; subsequently, the other terms come. For example, x3 \u2013 3x2 + x -12 is a standard polynomial. So the highest degree in the given polynomial expression is 3, then comes 2, and then 1. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Is_6_a_polynomial\"><\/span>Is 6 a polynomial?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\t6 can be written as 6x0 or 0x2+0x+6, representing the polynomial expression. Therefore, we can consider 6 as a polynomial. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is a Polynomial?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"A polynomial is an expression consisting of variables (or indeterminate), exponents, and constants. For example, 3x2 -2x-10 is a polynomial.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the standard form of the polynomial?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"A standard polynomial is one where the highest degree is the first term; subsequently, the other terms come. For example, x3 \u2013 3x2 + x -12 is a standard polynomial. So the highest degree in the given polynomial expression is 3, then comes 2, and then 1.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Is 6 a polynomial?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"6 can be written as 6x0 or 0x2+0x+6, representing the polynomial expression. Therefore, we can consider 6 as a polynomial.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Polynomials Polynomials are defined as the type of algebraic expressions whose variables have only non-negative integers as their [&hellip;]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Polynomials","_yoast_wpseo_title":"Polynomials - Definitions, Types, Properties and Notation of Polynomials","_yoast_wpseo_metadesc":"Polynomials is math expressions with variables and coefficients used to solve problems. 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