{"id":9149,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9149"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","slug":"concept-of-polynomial","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/concept-of-polynomial\/","title":{"rendered":"Concept Of Polynomial"},"content":{"rendered":"<p><strong>Unit: <\/strong><strong>Algebra<\/strong><\/p>\n<p><strong>Chapter: <\/strong><strong>Concept of Polynomials<\/strong><\/p>\n<p><em>Reference: &#8211; Introduction to Polynomials, Terms and Coefficients, Degree of a Polynomial, Types of Polynomials, Zeroes of a Polynomial, Remainder Theorem, Factor Theorem, Factorization of Polynomials, Algebraic Identities<\/em><\/p>\n<p><strong>After studying this chapter, you should be able to understand:<\/strong><\/p>\n<ul>\n<li>The fundamental definition and components of a polynomial.<\/li>\n<li>How to classify polynomials based on degree and number of terms.<\/li>\n<li>The relationship between zeroes and factors of a polynomial.<\/li>\n<li>The application of the Remainder Theorem and Factor Theorem.<\/li>\n<\/ul>\n<p><strong>Introduction to Polynomials<\/strong><\/p>\n<p><strong><u>Definition<\/u><\/strong><\/p>\n<p>A polynomial is an algebraic expression consisting of variables (also called indeterminates), coefficients, and non-negative integer exponents, combined using addition, subtraction, and multiplication operations. A polynomial in one variable, x, is generally written as:<br \/>\n<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/M0rGk71GQNXS1764938809.gif?time=1764938810\" width=\"352\" \/><\/p>\n<p>where&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/6HnV7h6f8qvn1764938833.gif?time=1764938834\" width=\"149\" \/>&nbsp;are constants (coefficients), and&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/zkt3TWMeVaeG1764938833.gif?time=1764938833\" width=\"12\" \/>&nbsp;is a non-negative integer.<\/p>\n<p><strong>[Importance of Polynomials]<\/strong><\/p>\n<ul>\n<li>Polynomials are the most basic and widely used algebraic expressions.<\/li>\n<li>They form the foundation for higher mathematics, including calculus and linear algebra.<\/li>\n<li>Used in various real-world applications, such as physics, engineering, and economics.<\/li>\n<li>Essential for solving equations and modeling situations.<\/li>\n<\/ul>\n<p><strong>Example<\/strong><\/p>\n<p><strong>Expression:<\/strong>&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/gfu2ZLahmpLT1764938833.gif?time=1764938834\" width=\"113\" \/><br \/>\nThis is a polynomial in x with three terms. The coefficients are 3, 2, and -5. The exponents are 2, 1, and 0.<\/p>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Components of a Polynomial<\/strong><\/p>\n<ul>\n<li><strong>Terms:<\/strong>&nbsp;Parts of the polynomial separated by + or &#8211; signs. E.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/xhs5HJcyrezf1764938827.gif?time=1764938827\" width=\"32\" \/>,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/Pq7ANzig4Zgg1764938827.gif?time=1764938827\" width=\"22\" \/>, and&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/fgqWBInxVtY21764938827.gif?time=1764938828\" width=\"26\" \/>&nbsp;are terms.<\/li>\n<li><strong>Coefficients:<\/strong>&nbsp;The numerical part of each term. E.g., 3, 2, and -5.<\/li>\n<li><strong>Variable:<\/strong>&nbsp;The symbol whose value can change. Commonly x, y, z.<\/li>\n<li><strong>Exponent:<\/strong>&nbsp;The power to which the variable is raised. Must be a non-negative integer.<\/li>\n<\/ul>\n<p><strong>Key Points:<\/strong><\/p>\n<ul>\n<li>Expressions with variables in the denominator (e.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"37\" src=\"https:\/\/app.kapdec.com\/questions-images\/44pSxlag8YH91764938827.gif?time=1764938828\" width=\"9\" \/>) or under a radical (e.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"29\" src=\"https:\/\/app.kapdec.com\/questions-images\/JDYwIfVciw9o1764938825.gif?time=1764938826\" width=\"24\" \/>) are not polynomials.<\/li>\n<li>The coefficient of the term with the highest exponent is called the leading coefficient.<\/li>\n<\/ul>\n<p><strong>2. Standard Form<\/strong><\/p>\n<p>A polynomial is written in standard form when its terms are arranged in descending order of their exponents.<\/p>\n<p><strong>Terms and Coefficients<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>This section delves deeper into the building blocks of polynomials. Understanding terms and coefficients is crucial for performing operations like addition, subtraction, and factorization.<\/p>\n<p><strong>Importance of Terms and Coefficients<\/strong><\/p>\n<ul>\n<li>Necessary for identifying like terms during simplification.<\/li>\n<li>Helps in determining the degree and leading term.<\/li>\n<li>Fundamental for evaluating polynomials for given values of the variable.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>In the polynomial&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/HUTmMfENN2SF1764938826.gif?time=1764938826\" width=\"122\" \/>:\n<ul style=\"list-style-type:circle\">\n<li>Terms:&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/o9mHoWZ1T7Fg1764938826.gif?time=1764938826\" width=\"32\" \/>,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/dPh4NWrJmmf41764938809.gif?time=1764938810\" width=\"47\" \/>,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/xXjuHs7el4qX1764938826.gif?time=1764938827\" width=\"11\" \/><\/li>\n<li>Coefficients: 4, -2, 7<\/li>\n<li>Constant term: 7 (the term with&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/9oawl1muXQma1764938828.gif?time=1764938829\" width=\"21\" \/>)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Like and Unlike Terms<\/strong><\/p>\n<ul>\n<li><strong>Like Terms:<\/strong>&nbsp;Terms that have the same variable raised to the same power. E.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/aWlUeuWz1vUH1764938828.gif?time=1764938829\" width=\"32\" \/>&nbsp;and&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/dGvbCAA415yg1764938828.gif?time=1764938829\" width=\"47\" \/>&nbsp;are like terms.<\/li>\n<li><strong>Unlike Terms:<\/strong>&nbsp;Terms with different variables or different exponents. E.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/BDOr45bBSt2o1764938828.gif?time=1764938829\" width=\"22\" \/>&nbsp;and&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/u82qGQako2pS1764938829.gif?time=1764938829\" width=\"23\" \/>&nbsp;are unlike;&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/vwiAKaGP8rK91764938823.gif?time=1764938824\" width=\"32\" \/>&nbsp;and&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/0OqwM29b2w2l1764938824.gif?time=1764938824\" width=\"22\" \/>&nbsp;are unlike.<\/li>\n<\/ul>\n<p><strong>2. Constant Polynomial<\/strong><\/p>\n<p>A polynomial of degree 0. It has no variable part and is just a constant number. E.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/xh7UWZ5s6P6x1764938824.gif?time=1764938824\" width=\"76\" \/>.<\/p>\n<p><strong>Degree of a Polynomial<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>The degree of a polynomial is the highest exponent of the variable in any of its terms when the polynomial is expressed in its standard form.<\/p>\n<p><strong>[Importance of Degree]<\/strong><\/p>\n<ul>\n<li>Determines the general shape and behavior of the polynomial&#39;s graph.<\/li>\n<li>Indicates the maximum number of zeroes (or roots) the polynomial can have.<\/li>\n<li>Used to classify polynomials (linear, quadratic, cubic, etc.).<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/4F9s0zcRN2lQ1764938821.gif?time=1764938822\" width=\"167\" \/>&nbsp;has a degree of 4.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/1L2Lo9jAZRuE1764938821.gif?time=1764938822\" width=\"77\" \/>&nbsp;has a degree of 0.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Finding the Degree<\/strong><\/p>\n<p>Identify the term with the largest exponent. The value of that exponent is the degree.<\/p>\n<p><strong>2. Degree of a Zero Polynomial<\/strong><\/p>\n<p>The polynomial&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/YCjsl9y6pmsH1764938822.gif?time=1764938822\" width=\"76\" \/>&nbsp;is called the zero polynomial. Its degree is not defined.<\/p>\n<p><strong>Types of Polynomials<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>Polynomials can be classified based on the number of terms they contain or based on their degree.<\/p>\n<p><strong>Importance of Classification<\/strong><\/p>\n<ul>\n<li>Helps in quickly identifying the properties of the polynomial.<\/li>\n<li>Different types have standard methods for solving and factoring.<\/li>\n<li>Aids in communication and problem-solving.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>Based on number of terms:\n<ul style=\"list-style-type:circle\">\n<li><strong>Monomial:<\/strong>&nbsp;One term (e.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/TYt5LxDmDH821764938822.gif?time=1764938822\" width=\"32\" \/>)<\/li>\n<li><strong>Binomial:<\/strong>&nbsp;Two terms (e.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/0vFxK45ZSzBl1764938822.gif?time=1764938823\" width=\"46\" \/>)<\/li>\n<li><strong>Trinomial:<\/strong>&nbsp;Three terms (e.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/S5tvyyEOgipi1764938822.gif?time=1764938823\" width=\"113\" \/>)<\/li>\n<\/ul>\n<\/li>\n<li>Based on degree:\n<ul style=\"list-style-type:circle\">\n<li><strong>Linear Polynomial:<\/strong>&nbsp;Degree 1 (e.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/mrARYfKAljKh1764938822.gif?time=1764938823\" width=\"57\" \/>)<\/li>\n<li><strong>Quadratic Polynomial:<\/strong>&nbsp;Degree 2 (e.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/BkVVOUQOS3Vc1764938823.gif?time=1764938823\" width=\"102\" \/>)<\/li>\n<li><strong>Cubic Polynomial:<\/strong>&nbsp;Degree 3 (e.g.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/LvpkV4x6txrr1764938823.gif?time=1764938823\" width=\"157\" \/>)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Based on Number of Terms<\/strong><\/p>\n<ul>\n<li><strong>Monomial, Binomial, Trinomial, Polynomial<\/strong>&nbsp;(for four or more terms).<\/li>\n<\/ul>\n<p><strong>2. Based on Degree<\/strong><\/p>\n<ul>\n<li><strong>Constant (Degree 0), Linear (Degree 1), Quadratic (Degree 2), Cubic (Degree 3), Quartic (Degree 4),<\/strong>&nbsp;and so on.<\/li>\n<\/ul>\n<p><strong>Zeroes of a Polynomial<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>A zero (or root) of a polynomial&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/bi1jPfl7Uuis1764938823.gif?time=1764938824\" width=\"39\" \/>&nbsp;is a number&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/sjVi7pr0k2T01764938819.gif?time=1764938819\" width=\"13\" \/>&nbsp;such that when it is substituted for the variable, the value of the polynomial becomes zero, i.e.,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/afpKxEuz0XnZ1764938819.gif?time=1764938820\" width=\"78\" \/>.<\/p>\n<p><strong>[Importance of Zeroes]<\/strong><\/p>\n<ul>\n<li>Finding zeroes is equivalent to solving the equation&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/yoSuVGsuhs5J1764938819.gif?time=1764938820\" width=\"76\" \/>.<\/li>\n<li>Zeroes represent the x-intercepts of the polynomial&#39;s graph.<\/li>\n<li>Directly related to the factors of the polynomial.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>For&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/UDC10XZrAZmn1764938819.gif?time=1764938820\" width=\"167\" \/>, find&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/uLIyyxRaIs1D1764938819.gif?time=1764938820\" width=\"194\" \/>. So, 2 is a zero.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Finding Zeroes<\/strong><\/p>\n<p>Set the polynomial equal to zero and solve for the variable. For linear and quadratic polynomials, this can be done directly.<\/p>\n<p><strong>2. Relationship with Factors<\/strong><\/p>\n<p>If&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/gZMJ0l3vyQIR1764938819.gif?time=1764938820\" width=\"13\" \/>&nbsp;is a zero of&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/40G248BjSoqa1764938820.gif?time=1764938820\" width=\"39\" \/>, then&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/W5D20S45bkzh1764938820.gif?time=1764938821\" width=\"64\" \/>&nbsp;is a factor of&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/U43DKm0HHK7E1764938820.gif?time=1764938821\" width=\"39\" \/>.<\/p>\n<p><strong>Remainder Theorem<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>The Remainder Theorem states that when a polynomial&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/gtd3aGsaKsrK1764938820.gif?time=1764938821\" width=\"39\" \/>&nbsp;is divided by a linear divisor of the form&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/aawLYXLo8bhN1764938820.gif?time=1764938821\" width=\"63\" \/>, the remainder is equal to&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/b8t5ZivfYJeA1764938817.gif?time=1764938818\" width=\"40\" \/>.<\/p>\n<p><strong>[Importance of Remainder Theorem]<\/strong><\/p>\n<ul>\n<li>Provides a quick way to find the remainder without performing long division.<\/li>\n<li>Useful for verifying factors.<\/li>\n<li>Helps in evaluating polynomials at specific points.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>Find the remainder when&nbsp;<em>p(x)=<\/em><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/cssw28C7TKzb1764938817.gif?time=1764938818\" width=\"212\" \/>&nbsp;is divided by&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/tTHQi1PSXc7g1764938817.gif?time=1764938818\" width=\"63\" \/>.<\/li>\n<li>By Remainder Theorem, remainder =&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/z1v1aSC2060U1764938818.gif?time=1764938818\" width=\"218\" \/>.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Statement and Proof<\/strong><\/p>\n<p>If&nbsp;<em>p(x)<\/em>&nbsp;is divided by&nbsp;<em>(x-a)<\/em>, then&nbsp;<em>p(x)=(x-a)q(x)+r<\/em>, where&nbsp;<em>r<\/em>&nbsp;is the remainder. Substituting&nbsp;<em>x=a<\/em>&nbsp;gives&nbsp;<em>p(a)=r<\/em>.<\/p>\n<p><strong>2. Application<\/strong><\/p>\n<p>Used to check if&nbsp;<em>(x-a)<\/em>&nbsp;is a factor. If&nbsp;<em>p(a)=0<\/em>, then it is a factor.<\/p>\n<p><strong>Factor Theorem<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>The Factor Theorem is a special case of the Remainder Theorem. It states that&nbsp;<em>(x-a)<\/em>&nbsp;is a factor of the polynomial&nbsp;<em>p(x)<\/em>&nbsp;if and only if&nbsp;<em>p(a)=0<\/em>.<\/p>\n<p><strong>[Importance of Factor Theorem]<\/strong><\/p>\n<ul>\n<li>A powerful tool for factorizing polynomials.<\/li>\n<li>Simplifies the process of finding all factors and zeroes.<\/li>\n<li>Essential for solving polynomial equations.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>Check if&nbsp;<em>(x-1)<\/em>&nbsp;is a factor of&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/Y7MSsjMO5SlP1764938814.gif?time=1764938814\" width=\"223\" \/>.<\/li>\n<li><em>p(1)=1-3+3-1=0<\/em>. Yes, it is a factor.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Statement and Proof<\/strong><\/p>\n<p>Direct consequence of the Remainder Theorem. If&nbsp;<em>p(a)=0<\/em>, then remainder is 0, so&nbsp;<em>(x-a)<\/em>&nbsp;divides&nbsp;<em>p(x)<\/em>&nbsp;exactly.<\/p>\n<p><strong>2. Finding Factors<\/strong><\/p>\n<p>Use the Factor Theorem to test possible values of &#39;a&#39; (often factors of the constant term) to find zeroes and thus factors.<\/p>\n<p><strong>Factorization of Polynomials<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>Factorization is the process of expressing a polynomial as a product of its linear or irreducible factors. This is often done by finding the zeroes of the polynomial.<\/p>\n<p><strong>[Importance of Factorization]<\/strong><\/p>\n<ul>\n<li>Simplifies polynomial expressions.<\/li>\n<li>Essential for solving polynomial equations.<\/li>\n<li>Used in calculus for integration and finding limits.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>Factorize&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/Fb8mUPRDRl6U1764938815.gif?time=1764938815\" width=\"102\" \/>.<\/li>\n<li>The zeroes are 2 and 3, so factors are&nbsp;<em>(x-2)<\/em>&nbsp;and&nbsp;<em>(x-3)<\/em>. Thus,&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/lxiGh4uVtN5x1764938815.gif?time=1764938816\" width=\"253\" \/>.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. By Splitting the Middle Term<\/strong><\/p>\n<p>A common method for quadratic polynomials.<\/p>\n<p><strong>2. Using Factor Theorem<\/strong><\/p>\n<p>For higher-degree polynomials, use the Factor Theorem to find one factor, then perform polynomial division to reduce the degree.<\/p>\n<p><strong>Algebraic Identities<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>Algebraic identities are standard equations that are true for all values of the variables involved. They are useful shortcuts for expanding and factorizing polynomials.<\/p>\n<p><strong>[Importance of Algebraic Identities]<\/strong><\/p>\n<ul>\n<li>Speed up calculations and simplifications.<\/li>\n<li>Provide standard forms for factorization.<\/li>\n<li>Frequently used in problem-solving.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/hi16oIwcUemn1764938810.gif?time=1764938811\" width=\"221\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/a7sRJfuLKXxO1764938811.gif?time=1764938811\" width=\"221\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/Mt5kVSi3BgnI1764938811.gif?time=1764938811\" width=\"218\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/IjVlykYYbZeu1764938811.gif?time=1764938812\" width=\"284\" \/><\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Common Identities<\/strong><\/p>\n<p>Memorizing key identities is crucial for efficient problem-solving.<\/p>\n<p><strong>2. Application in Factorization<\/strong><\/p>\n<p>Recognizing patterns that match these identities allows for quick factorization.<\/p>\n<p><strong>[Example: -]<\/strong><\/p>\n<p>Consider the polynomial&nbsp;<em>p(x)=<\/em><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/kPt5ev8OFh0W1764938811.gif?time=1764938812\" width=\"245\" \/>.<\/p>\n<p><strong>Question:<\/strong><br \/>\na) Find the degree of the polynomial and identify its type based on the number of terms.<br \/>\nb) Verify whether&nbsp;<em>(x-2)<\/em>&nbsp;is a factor of&nbsp;<em>p(x)<\/em>&nbsp;using the Factor Theorem.<br \/>\nc) If it is a factor, factorize&nbsp;<em>p(x)<\/em>&nbsp;completely.<br \/>\nd) Find all the zeroes of&nbsp;<em>p(x)<\/em>.<\/p>\n<p>Prove your answer by providing a step-by-step solution and giving&nbsp;<strong>three independent reasons<\/strong>&nbsp;supporting your conclusion for part (b) from these domains:&nbsp;<strong>(A) Direct Substitution, (B) Remainder Theorem Application, (C) Polynomial Long Division Verification.<\/strong><\/p>\n<p><strong>[Solution: -]<\/strong><\/p>\n<p><strong>a) Degree and Type<\/strong><\/p>\n<ul>\n<li>The highest power of x is 3. So, the&nbsp;<strong>degree<\/strong>&nbsp;is 3.<\/li>\n<li>The polynomial has 4 terms:&nbsp;<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/NLWbnidkxlFw1764938809.gif?time=1764938810\" width=\"175\" \/>. Therefore, it is simply called a&nbsp;<strong>polynomial<\/strong>&nbsp;(or specifically, a cubic polynomial).<\/li>\n<\/ul>\n<p><strong>b) Verify if&nbsp;<\/strong><em>(x-2)<\/em><strong>&nbsp;is a factor using the Factor Theorem.<\/strong><\/p>\n<p>The Factor Theorem states that&nbsp;<em>(x-a)<\/em>&nbsp;is a factor of&nbsp;<em>p(x)<\/em>&nbsp;if and only if&nbsp;<em>p(a)=0<\/em>. Here,&nbsp;<em>a=2<\/em>.<\/p>\n<p><strong>(A) Direct Substitution<\/strong><br \/>\nCompute&nbsp;<em>p(2)<\/em>:<br \/>\n<em>=<\/em><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/SdOzHgBlGw7u1764938812.gif?time=1764938813\" width=\"293\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/663K4H7c70Ws1764938812.gif?time=1764938813\" width=\"203\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/u0epaLZ5SRUt1764938812.gif?time=1764938813\" width=\"170\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/AiJmYDKaK0tC1764938813.gif?time=1764938813\" width=\"218\" \/><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/ZD56uGpwQA561764938813.gif?time=1764938813\" width=\"152\" \/><\/p>\n<p>Since&nbsp;<em>p(2)=-12<\/em>&ne;<em>0<\/em>, by the Factor Theorem,&nbsp;<em>(x-2)<\/em>&nbsp;is&nbsp;<strong>not<\/strong>&nbsp;a factor.<\/p>\n<p><strong>(B) Remainder Theorem Application<\/strong><br \/>\nThe Remainder Theorem states that the remainder when&nbsp;<em>p(x)<\/em>&nbsp;is divided by&nbsp;<em>(x-2)<\/em>&nbsp;is&nbsp;<em>p(2)<\/em>. We calculated&nbsp;<em>p(2)=-12<\/em>. A non-zero remainder means that&nbsp;<em>(x-2)<\/em>&nbsp;does not divide&nbsp;<em>p(x)<\/em>&nbsp;exactly. Therefore, it is&nbsp;<strong>not<\/strong>&nbsp;a factor. This is a direct application of the theorem and is consistent with (A).<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Unit: Algebra Chapter: Concept of Polynomials Reference: &#8211; Introduction to Polynomials, Terms and Coefficients, Degree of a Polynomial, Types of Polynomials, Zeroes of a Polynomial, Remainder Theorem, Factor Theorem, Factorization of Polynomials, Algebraic Identities After studying this chapter, you should be able to understand: The fundamental definition and components of a polynomial. How to classify [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[570],"tags":[],"class_list":["post-9149","post","type-post","status-publish","format-standard","hentry","category-math-sci-olympiad"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9149","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9149"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9149\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9149"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9149"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}