{"id":27135,"date":"2021-03-28T13:00:35","date_gmt":"2021-03-28T13:00:35","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27135"},"modified":"2024-08-18T07:23:46","modified_gmt":"2024-08-18T07:23:46","slug":"algebra-of-polynomials-of-real-numbers","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/algebra-of-polynomials-of-real-numbers\/","title":{"rendered":"Algebra of Polynomials of Real Numbers"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px; text-align:center;\">\n<h1>Polynomial Algebra of Real Numbers<\/h1>\n<p>    <em><strong>Summary:<\/strong><br \/>\n        In this class, we will explore polynomial algebra, its definition, properties, and applications. Polynomials are a fundamental part of mathematics and have broad applications in various disciplines.<br \/>\n    <\/em><\/p>\n<p>    <strong>LEARNING OBJECTIVES<\/strong><\/p>\n<p>By the end of this class, the student will be able to:<\/p>\n<p style=\"text-align:left;\">\n        1. Define and understand polynomials and their properties.<br \/>\n        2. Identify the degree and coefficients of a polynomial.<br \/>\n        3. Perform algebraic operations with polynomials and apply their properties in mathematical contexts.\n    <\/p>\n<p>    <strong>CONTENT INDEX:<\/strong><\/p>\n<p>\n        <a href=\"#1\"><strong>1. Polynomial Algebra: Definitions<\/strong><\/a><br \/>\n        <a href=\"#2\"><strong>2. Types of Polynomials<\/strong><\/a><br \/>\n        <a href=\"#3\"><strong>3. Polynomial Algebra: Operations<\/strong><\/a><br \/>\n        <a href=\"#4\"><strong>4. Factorization and Division of Polynomials<\/strong><\/a>\n    <\/p>\n<p>    <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ry4sKaS3RMc\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2><strong>1. Polynomial Algebra: Definitions<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=139s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">To understand Polynomial Algebra, we must first know what polynomials are.<\/span><\/strong><\/a> Polynomials are algebraic functions. If <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> is a real variable, then the function <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> is called a polynomial if it can be written in the form:\n<\/p>\n<p style=\"text-align: center; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(x)= \\sum_{i=0}^n a_i x^i= a_0 + a_1x + a_2x^2 + a_3x^3 + \\cdots + a_nx^n,<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    where <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> is a non-negative integer, and all <span class=\"katex-eq\" data-katex-display=\"false\">a_i<\/span>, with <span class=\"katex-eq\" data-katex-display=\"false\">i\\in\\{1,2,3,\\cdots,n\\},<\/span> are real coefficients. If there is a <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> such that <span class=\"katex-eq\" data-katex-display=\"false\">a_k\\neq 0<\/span> and, when <span class=\"katex-eq\" data-katex-display=\"false\">k\\lt i<\/span>, it follows that <span class=\"katex-eq\" data-katex-display=\"false\">a_i=0<\/span>, then such a value of <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> is called <strong>the degree of the polynomial.<\/strong> In other words, the degree of a polynomial is the highest power that accompanies a non-zero coefficient.\n<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2><strong>2. Types of Polynomials<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=340s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Polynomials are classified according to their degree;<\/span><\/strong><\/a> therefore, when mentioning a polynomial, it is almost always stated that it is a polynomial of degree <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span>, where <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> is the highest power of <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> that accompanies the non-zero coefficient of such a polynomial.\n<\/p>\n<h3>2.1. Constant Polynomials<\/h3>\n<p style=\"text-align: justify; color: #000000;\">This is the family that includes all zero-degree polynomials and the null polynomial. We say that a polynomial is of degree zero if it can be written in the form <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=c,<\/span> with <span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0.<\/span> On the other hand, the null polynomial is of the form <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 0<\/span>, and for this, a degree is not defined.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2><strong>3. Polynomial Algebra: Operations<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=428s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Polynomials inherit all their properties from the algebra of real numbers.<\/span><\/strong><\/a> Distributive and associative properties are especially relevant.\n<\/p>\n<h3>3.1. Addition and Subtraction<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=470s\" target=\"_blank\" rel=\"noopener\"> <strong><span style=\"color: #ff0000;\">If <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> are two polynomials of degree<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">m<\/span>, respectively, with<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">m=n+k<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">0\\leq k,<\/span>\n<p style=\"text-align: justify; color: #000000;\">then it follows that:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle P(x) \\pm Q(x) &amp;=\\displaystyle \\sum_{i=0}^n a_i x^i \\pm \\sum_{i=0}^m b_i x^i \\\\ \\\\\n\n &amp;\\displaystyle = \\sum_{i=0}^n a_i x^i \\pm \\left( \\sum_{i=0}^n b_i x^i + \\sum_{i=n+1}^{n+k} b_i x^i \\right) \\\\ \\\\\n\n&amp;\\displaystyle = \\sum_{i=0}^n (a_i \\pm b_i) x^i + \\sum_{i=n+1}^m b_i x^i\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: justify; color: #000000;\">That is, the coefficients that accompany equal powers of <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> are added or subtracted, as appropriate.<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\">EXAMPLE:<\/span><br \/>\nIf <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 3+5x+2x^2<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 6x-3x^2 +23x^5<\/span>, then:<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">P(x) + Q(x) = \\cdots \\\\ = (3+5x+2x^2) + (6x-3x^2 +23x^5) \\\\ = 3 + (5+6)x + (2-3)x^2 + 23x^5 \\\\ = 3 + 11x - x^2 + 23x^5 <\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">P(x) - Q(x) = \\cdots \\\\ = (3+5x+2x^2) - (6x-3x^2 +23x^5) \\\\ = 3 + (5-6)x + (2+3)x^2 - 23x^5 \\\\ = 3 - x + 5x^2 - 23x^5 <\/span>\n<\/p>\n<h3>3.2. Multiplication<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=894s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">In the same context as for the addition and subtraction of polynomials,<\/span><\/strong><\/a> the product of polynomials will be developed as follows:\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    First, we distinguish scalar multiplication. If <span class=\"katex-eq\" data-katex-display=\"false\">c \\in \\mathbb{R},<\/span> then we have:\n<\/p>\n<p style=\"text-align: center; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle c P(x) = c \\sum_{i=0}^n a_i x^i =\\sum_{i=0}^n c a_i x^i <\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    And then we have the multiplication of polynomials:\n<\/p>\n<p style=\"text-align: center; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\displaystyle P(x) Q(x) &amp;\\displaystyle = \\left( \\sum_{i=0}^n a_i x^i \\right) \\left(\\sum_{j=0}^m b_j x^j\\right) \\\\ \\\\\n\n&amp;=\\displaystyle \\left[\\sum_{j=0}^m \\left( \\sum_{i=0}^n a_i x^i \\right) b_j x^j\\right] \\\\ \\\\\n\n&amp;=\\displaystyle \\sum_{j=0}^m \\left( \\sum_{i=0}^n a_ib_j x^{i+j} \\right) \\\\ \\\\\n\n&amp;=\\displaystyle \\sum_{i,j=0}^{n,m} a_ib_j x^{i+j}\n\n\\end{array}<\/span>\n<p style=\"text-align: justify; color: #000000;\">\n    This is what we would summarize as \u00abthe sum of the products of all with all.\u00bb\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span style=\"color: #000080;\">EXAMPLE:<\/span><br \/>\n    If <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 4x+ 2x^2-x^4<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 5 - x + x^2-7x^3,<\/span> then:\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">P(x)Q(x) =\\cdots \\\\ {} \\\\= (4x+ 2x^2-x^4)(5 - x + x^2-7x^3) \\\\ {} \\\\ = 4x(5 - x + x^2-7x^3) \\\\ + 2x^2 (5 - x + x^2-7x^3) \\\\ - x^4 (5 - x + x^2-7x^3) \\\\ {} \\\\ = 20x - 4x^2 + 4x^3 - 28x^4 \\\\ + 10x^2 - 2x^3 + 2x^4 - 14x^5 \\\\ -5x^4 + x^5 - x^6 + 7x^7 \\\\ {} \\\\ = 20x + 6x^2 + 2x^3 - 31x^4 - 13x^5 - x^6 + 7x^7<\/span>\n<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2><strong>4. Factorization and Division of Polynomials<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1375s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">When we multiply two polynomials, we transform two simple polynomials into a more complex one (of higher degree).<\/span><\/strong><\/a> When we factorize a polynomial, we follow the inverse process: we transform a complex polynomial into the product of two or more lower-degree polynomials.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    To factorize a polynomial <span class=\"katex-eq\" data-katex-display=\"false\">P(x),<\/span> it is necessary to find the values of <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> that nullify the polynomial; if such values exist, then the polynomial is factorable. Talking about existence is straightforward, but finding them is a different story. We will review this topic in more detail when we study the factorizations of quadratic and (2n) quadratic polynomials.\n<\/p>\n<h3>4.1. Notable Products<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1654s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">There are, however, cases where factorization is easily obtained,<\/span><\/strong><br \/>\n    <\/a> as in the case of notable products. Some of these results are as follows:\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">x^2 - y^2 = (x-y)(x+y)<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">(x\\pm y)^2 = x^2 \\pm 2xy + y^2<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">(x \\pm y)^3 = x^3 \\pm 3x^2y + 3xy^2 \\pm y^3<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">x^3-y^3=(x-y)(x^2+xy+y^2)<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">x^3+y^3=(x+y)(x^2-xy+y^2)<\/span>\n<\/p>\n<h3>4.2. The Division Algorithm<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1854s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Just as multiplying integers gives us composite numbers, and division through the division algorithm allows us to factorize when the remainder is zero,<\/span><\/strong><\/a> the same happens with polynomials. Explaining the division algorithm \u00abin text\u00bb can be a bit complicated; it&#8217;s much easier to understand by directly seeing how it&#8217;s done and in which cases the algorithm leads to factorization. To achieve this, we will review some examples.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span style=\"color: #000080;\">EXAMPLE:<\/span> Calculate <span class=\"katex-eq\" data-katex-display=\"false\">P(x):Q(x)<\/span> for the following cases:\n<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=2 x^3 + x^2 - 2 x - 1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x-1<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1930s\" target=\"_blank\" rel=\"noopener\"> <strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong> <\/a>\n    <\/li>\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=x^4+2x^3-x+1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x^2-4<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=2120s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span> <\/a>\n    <\/li>\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=3 x^4 - 2 x^3 - x^2 - 4 x + 1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x^2+x+1<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=2331s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span> <\/a>\n    <\/li>\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=x^7+5x^4+5x^2-3x+1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x^3-2x^2+1<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=2464s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span> <\/a>\n    <\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Polynomial Algebra of Real Numbers Summary: In this class, we will explore polynomial algebra, its definition, properties, and applications. Polynomials are a fundamental part of mathematics and have broad applications in various disciplines. LEARNING OBJECTIVES By the end of this class, the student will be able to: 1. Define and understand polynomials and their properties. 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