{"id":9925,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9925"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"concept-of-polynomial","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/concept-of-polynomial\/","title":{"rendered":"Concept Of Polynomial"},"content":{"rendered":"
\n
\n

KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

\n

Unit: <\/strong>Algebra<\/strong><\/p>\n

Chapter: <\/strong>Concept of Polynomials<\/strong><\/p>\n

Reference: – 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

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • The fundamental definition and components of a polynomial.<\/li>\n
  • How to classify polynomials based on degree and number of terms.<\/li>\n
  • The relationship between zeroes and factors of a polynomial.<\/li>\n
  • The application of the Remainder Theorem and Factor Theorem.<\/li>\n<\/ul>\n

    Introduction to Polynomials<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    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:<\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    where\u00a0<\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u00a0are constants (coefficients), and\u00a0<\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u00a0is a non-negative integer.<\/p>\n

    [Importance of Polynomials]<\/strong><\/p>\n

      \n
    • Polynomials are the most basic and widely used algebraic expressions.<\/li>\n
    • They form the foundation for higher mathematics, including calculus and linear algebra.<\/li>\n
    • Used in various real-world applications, such as physics, engineering, and economics.<\/li>\n
    • Essential for solving equations and modeling situations.<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      Expression:<\/strong>\u00a0<\/p>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      \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

      [Subtopics]<\/strong><\/p>\n

      1. Components of a Polynomial<\/strong><\/p>\n

        \n
      • Terms:<\/strong>\u00a0Parts of the polynomial separated by + or – signs. E.g.,\u00a0\n
        \"\"<\/p>\n

        Source: Kapdec.com<\/p>\n<\/div>\n

        ,\u00a0<\/p>\n

        \"\"<\/p>\n

        Source: Kapdec.com<\/p>\n<\/div>\n

        , and\u00a0<\/p>\n

        \"\"<\/p>\n

        Source: Kapdec.com<\/p>\n<\/div>\n

        \u00a0are terms.<\/li>\n

      • Coefficients:<\/strong>\u00a0The numerical part of each term. E.g., 3, 2, and -5.<\/li>\n
      • Variable:<\/strong>\u00a0The symbol whose value can change. Commonly x, y, z.<\/li>\n
      • Exponent:<\/strong>\u00a0The power to which the variable is raised. Must be a non-negative integer.<\/li>\n<\/ul>\n

        Key Points:<\/strong><\/p>\n

          \n
        • Expressions with variables in the denominator (e.g.,\u00a0\n
          \"\"<\/p>\n

          Source: Kapdec.com<\/p>\n<\/div>\n

          ) or under a radical (e.g.,\u00a0<\/p>\n

          \"\"<\/p>\n

          Source: Kapdec.com<\/p>\n<\/div>\n

          ) are not polynomials.<\/li>\n

        • The coefficient of the term with the highest exponent is called the leading coefficient.<\/li>\n<\/ul>\n

          2. Standard Form<\/strong><\/p>\n

          A polynomial is written in standard form when its terms are arranged in descending order of their exponents.<\/p>\n

          Terms and Coefficients<\/strong><\/p>\n

          [Definition]<\/strong><\/p>\n

          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

          Importance of Terms and Coefficients<\/strong><\/p>\n

            \n
          • Necessary for identifying like terms during simplification.<\/li>\n
          • Helps in determining the degree and leading term.<\/li>\n
          • Fundamental for evaluating polynomials for given values of the variable.<\/li>\n<\/ul>\n

            Examples<\/strong><\/p>\n

              \n
            • In the polynomial\u00a0\n
              \"\"<\/p>\n

              Source: Kapdec.com<\/p>\n<\/div>\n

              :<\/p>\n

                \n
              • Terms:\u00a0\n
                \"\"<\/p>\n

                Source: Kapdec.com<\/p>\n<\/div>\n

                ,\u00a0<\/p>\n

                \"\"<\/p>\n

                Source: Kapdec.com<\/p>\n<\/div>\n

                ,\u00a0<\/p>\n

                \"\"<\/p>\n

                Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n

              • Coefficients: 4, -2, 7<\/li>\n
              • Constant term: 7 (the term with\u00a0\n
                \"\"<\/p>\n

                Source: Kapdec.com<\/p>\n<\/div>\n

                )<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                [Subtopics]<\/strong><\/p>\n

                1. Like and Unlike Terms<\/strong><\/p>\n

                  \n
                • Like Terms:<\/strong>\u00a0Terms that have the same variable raised to the same power. E.g.,\u00a0\n
                  \"\"<\/p>\n

                  Source: Kapdec.com<\/p>\n<\/div>\n

                  \u00a0and\u00a0<\/p>\n

                  \"\"<\/p>\n

                  Source: Kapdec.com<\/p>\n<\/div>\n

                  \u00a0are like terms.<\/li>\n

                • Unlike Terms:<\/strong>\u00a0Terms with different variables or different exponents. E.g.,\u00a0\n
                  \"\"<\/p>\n

                  Source: Kapdec.com<\/p>\n<\/div>\n

                  \u00a0and\u00a0<\/p>\n

                  \"\"<\/p>\n

                  Source: Kapdec.com<\/p>\n<\/div>\n

                  \u00a0are unlike;\u00a0<\/p>\n

                  \"\"<\/p>\n

                  Source: Kapdec.com<\/p>\n<\/div>\n

                  \u00a0and\u00a0<\/p>\n

                  \"\"<\/p>\n

                  Source: Kapdec.com<\/p>\n<\/div>\n

                  \u00a0are unlike.<\/li>\n<\/ul>\n

                  2. Constant Polynomial<\/strong><\/p>\n

                  A polynomial of degree 0. It has no variable part and is just a constant number. E.g.,\u00a0<\/p>\n

                  \"\"<\/p>\n

                  Source: Kapdec.com<\/p>\n<\/div>\n

                  .<\/p>\n

                  Degree of a Polynomial<\/strong><\/p>\n

                  [Definition]<\/strong><\/p>\n

                  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

                  [Importance of Degree]<\/strong><\/p>\n

                    \n
                  • Determines the general shape and behavior of the polynomial’s graph.<\/li>\n
                  • Indicates the maximum number of zeroes (or roots) the polynomial can have.<\/li>\n
                  • Used to classify polynomials (linear, quadratic, cubic, etc.).<\/li>\n<\/ul>\n

                    Examples<\/strong><\/p>\n

                      \n
                    • \n
                      \"\"<\/p>\n

                      Source: Kapdec.com<\/p>\n<\/div>\n

                      \u00a0has a degree of 4.<\/li>\n

                    • \n
                      \"\"<\/p>\n

                      Source: Kapdec.com<\/p>\n<\/div>\n

                      \u00a0has a degree of 0.<\/li>\n<\/ul>\n

                      [Subtopics]<\/strong><\/p>\n

                      1. Finding the Degree<\/strong><\/p>\n

                      Identify the term with the largest exponent. The value of that exponent is the degree.<\/p>\n

                      2. Degree of a Zero Polynomial<\/strong><\/p>\n

                      The polynomial\u00a0<\/p>\n

                      \"\"<\/p>\n

                      Source: Kapdec.com<\/p>\n<\/div>\n

                      \u00a0is called the zero polynomial. Its degree is not defined.<\/p>\n

                      Types of Polynomials<\/strong><\/p>\n

                      [Definition]<\/strong><\/p>\n

                      Polynomials can be classified based on the number of terms they contain or based on their degree.<\/p>\n

                      Importance of Classification<\/strong><\/p>\n

                        \n
                      • Helps in quickly identifying the properties of the polynomial.<\/li>\n
                      • Different types have standard methods for solving and factoring.<\/li>\n
                      • Aids in communication and problem-solving.<\/li>\n<\/ul>\n

                        Examples<\/strong><\/p>\n

                          \n
                        • Based on number of terms:\n
                            \n
                          • Monomial:<\/strong>\u00a0One term (e.g.,\u00a0\n
                            \"\"<\/p>\n

                            Source: Kapdec.com<\/p>\n<\/div>\n

                            )<\/li>\n

                          • Binomial:<\/strong>\u00a0Two terms (e.g.,\u00a0\n
                            \"\"<\/p>\n

                            Source: Kapdec.com<\/p>\n<\/div>\n

                            )<\/li>\n

                          • Trinomial:<\/strong>\u00a0Three terms (e.g.,\u00a0\n
                            \"\"<\/p>\n

                            Source: Kapdec.com<\/p>\n<\/div>\n

                            )<\/li>\n<\/ul>\n<\/li>\n

                          • Based on degree:\n
                              \n
                            • Linear Polynomial:<\/strong>\u00a0Degree 1 (e.g.,\u00a0\n
                              \"\"<\/p>\n

                              Source: Kapdec.com<\/p>\n<\/div>\n

                              )<\/li>\n

                            • Quadratic Polynomial:<\/strong>\u00a0Degree 2 (e.g.,\u00a0\n
                              \"\"<\/p>\n

                              Source: Kapdec.com<\/p>\n<\/div>\n

                              )<\/li>\n

                            • Cubic Polynomial:<\/strong>\u00a0Degree 3 (e.g.,\u00a0\n
                              \"\"<\/p>\n

                              Source: Kapdec.com<\/p>\n<\/div>\n

                              )<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                              [Subtopics]<\/strong><\/p>\n

                              1. Based on Number of Terms<\/strong><\/p>\n

                                \n
                              • Monomial, Binomial, Trinomial, Polynomial<\/strong>\u00a0(for four or more terms).<\/li>\n<\/ul>\n

                                2. Based on Degree<\/strong><\/p>\n

                                  \n
                                • Constant (Degree 0), Linear (Degree 1), Quadratic (Degree 2), Cubic (Degree 3), Quartic (Degree 4),<\/strong>\u00a0and so on.<\/li>\n<\/ul>\n

                                  Zeroes of a Polynomial<\/strong><\/p>\n

                                  [Definition]<\/strong><\/p>\n

                                  A zero (or root) of a polynomial\u00a0<\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  \u00a0is a number\u00a0<\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  \u00a0such that when it is substituted for the variable, the value of the polynomial becomes zero, i.e.,\u00a0<\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  .<\/p>\n

                                  [Importance of Zeroes]<\/strong><\/p>\n

                                    \n
                                  • Finding zeroes is equivalent to solving the equation\u00a0\n
                                    \"\"<\/p>\n

                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                    .<\/li>\n

                                  • Zeroes represent the x-intercepts of the polynomial’s graph.<\/li>\n
                                  • Directly related to the factors of the polynomial.<\/li>\n<\/ul>\n

                                    Examples<\/strong><\/p>\n

                                      \n
                                    • For\u00a0\n
                                      \"\"<\/p>\n

                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                      , find\u00a0<\/p>\n

                                      \"\"<\/p>\n

                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                      . So, 2 is a zero.<\/li>\n<\/ul>\n

                                      [Subtopics]<\/strong><\/p>\n

                                      1. Finding Zeroes<\/strong><\/p>\n

                                      Set the polynomial equal to zero and solve for the variable. For linear and quadratic polynomials, this can be done directly.<\/p>\n

                                      2. Relationship with Factors<\/strong><\/p>\n

                                      If\u00a0<\/p>\n

                                      \"\"<\/p>\n

                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                      \u00a0is a zero of\u00a0<\/p>\n

                                      \"\"<\/p>\n

                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                      , then\u00a0<\/p>\n

                                      \"\"<\/p>\n

                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                      \u00a0is a factor of\u00a0<\/p>\n

                                      \"\"<\/p>\n

                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                      .<\/p>\n

                                      Remainder Theorem<\/strong><\/p>\n

                                      [Definition]<\/strong><\/p>\n

                                      The Remainder Theorem states that when a polynomial\u00a0<\/p>\n

                                      \"\"<\/p>\n

                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                      \u00a0is divided by a linear divisor of the form\u00a0<\/p>\n

                                      \"\"<\/p>\n

                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                      , the remainder is equal to\u00a0<\/p>\n

                                      \"\"<\/p>\n

                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                      .<\/p>\n

                                      [Importance of Remainder Theorem]<\/strong><\/p>\n

                                        \n
                                      • Provides a quick way to find the remainder without performing long division.<\/li>\n
                                      • Useful for verifying factors.<\/li>\n
                                      • Helps in evaluating polynomials at specific points.<\/li>\n<\/ul>\n

                                        Examples<\/strong><\/p>\n

                                          \n
                                        • Find the remainder when\u00a0p(x)=<\/em>\n
                                          \"\"<\/p>\n

                                          Source: Kapdec.com<\/p>\n<\/div>\n

                                          \u00a0is divided by\u00a0<\/p>\n

                                          \"\"<\/p>\n

                                          Source: Kapdec.com<\/p>\n<\/div>\n

                                          .<\/li>\n

                                        • By Remainder Theorem, remainder =\u00a0\n
                                          \"\"<\/p>\n

                                          Source: Kapdec.com<\/p>\n<\/div>\n

                                          .<\/li>\n<\/ul>\n

                                          [Subtopics]<\/strong><\/p>\n

                                          1. Statement and Proof<\/strong><\/p>\n

                                          If\u00a0p(x)<\/em>\u00a0is divided by\u00a0(x-a)<\/em>, then\u00a0p(x)=(x-a)q(x)+r<\/em>, where\u00a0r<\/em>\u00a0is the remainder. Substituting\u00a0x=a<\/em>\u00a0gives\u00a0p(a)=r<\/em>.<\/p>\n

                                          2. Application<\/strong><\/p>\n

                                          Used to check if\u00a0(x-a)<\/em>\u00a0is a factor. If\u00a0p(a)=0<\/em>, then it is a factor.<\/p>\n

                                          Factor Theorem<\/strong><\/p>\n

                                          [Definition]<\/strong><\/p>\n

                                          The Factor Theorem is a special case of the Remainder Theorem. It states that\u00a0(x-a)<\/em>\u00a0is a factor of the polynomial\u00a0p(x)<\/em>\u00a0if and only if\u00a0p(a)=0<\/em>.<\/p>\n

                                          [Importance of Factor Theorem]<\/strong><\/p>\n

                                            \n
                                          • A powerful tool for factorizing polynomials.<\/li>\n
                                          • Simplifies the process of finding all factors and zeroes.<\/li>\n
                                          • Essential for solving polynomial equations.<\/li>\n<\/ul>\n

                                            Examples<\/strong><\/p>\n

                                              \n
                                            • Check if\u00a0(x-1)<\/em>\u00a0is a factor of\u00a0\n
                                              \"\"<\/p>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              .<\/li>\n

                                            • p(1)=1-3+3-1=0<\/em>. Yes, it is a factor.<\/li>\n<\/ul>\n

                                              [Subtopics]<\/strong><\/p>\n

                                              1. Statement and Proof<\/strong><\/p>\n

                                              Direct consequence of the Remainder Theorem. If\u00a0p(a)=0<\/em>, then remainder is 0, so\u00a0(x-a)<\/em>\u00a0divides\u00a0p(x)<\/em>\u00a0exactly.<\/p>\n

                                              2. Finding Factors<\/strong><\/p>\n

                                              Use the Factor Theorem to test possible values of ‘a’ (often factors of the constant term) to find zeroes and thus factors.<\/p>\n

                                              Factorization of Polynomials<\/strong><\/p>\n

                                              [Definition]<\/strong><\/p>\n

                                              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

                                              [Importance of Factorization]<\/strong><\/p>\n

                                                \n
                                              • Simplifies polynomial expressions.<\/li>\n
                                              • Essential for solving polynomial equations.<\/li>\n
                                              • Used in calculus for integration and finding limits.<\/li>\n<\/ul>\n

                                                Examples<\/strong><\/p>\n

                                                  \n
                                                • Factorize\u00a0\n
                                                  \"\"<\/p>\n

                                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                                  .<\/li>\n

                                                • The zeroes are 2 and 3, so factors are\u00a0(x-2)<\/em>\u00a0and\u00a0(x-3)<\/em>. Thus,\u00a0\n
                                                  \"\"<\/p>\n

                                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                                  .<\/li>\n<\/ul>\n

                                                  [Subtopics]<\/strong><\/p>\n

                                                  1. By Splitting the Middle Term<\/strong><\/p>\n

                                                  A common method for quadratic polynomials.<\/p>\n

                                                  2. Using Factor Theorem<\/strong><\/p>\n

                                                  For higher-degree polynomials, use the Factor Theorem to find one factor, then perform polynomial division to reduce the degree.<\/p>\n

                                                  Algebraic Identities<\/strong><\/p>\n

                                                  [Definition]<\/strong><\/p>\n

                                                  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

                                                  [Importance of Algebraic Identities]<\/strong><\/p>\n

                                                    \n
                                                  • Speed up calculations and simplifications.<\/li>\n
                                                  • Provide standard forms for factorization.<\/li>\n
                                                  • Frequently used in problem-solving.<\/li>\n<\/ul>\n

                                                    Examples<\/strong><\/p>\n

                                                      \n
                                                    • \n
                                                      \"\"<\/p>\n

                                                      Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n

                                                    • \n
                                                      \"\"<\/p>\n

                                                      Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n

                                                    • \n
                                                      \"\"<\/p>\n

                                                      Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n

                                                    • \n
                                                      \"\"<\/p>\n

                                                      Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n<\/ul>\n

                                                      [Subtopics]<\/strong><\/p>\n

                                                      1. Common Identities<\/strong><\/p>\n

                                                      Memorizing key identities is crucial for efficient problem-solving.<\/p>\n

                                                      2. Application in Factorization<\/strong><\/p>\n

                                                      Recognizing patterns that match these identities allows for quick factorization.<\/p>\n

                                                      [Example: -]<\/strong><\/p>\n

                                                      Consider the polynomial\u00a0p(x)=<\/em><\/p>\n

                                                      \"\"<\/p>\n

                                                      Source: Kapdec.com<\/p>\n<\/div>\n

                                                      .<\/p>\n

                                                      Question:<\/strong>
                                                      \na) Find the degree of the polynomial and identify its type based on the number of terms.
                                                      \nb) Verify whether\u00a0(x-2)<\/em>\u00a0is a factor of\u00a0p(x)<\/em>\u00a0using the Factor Theorem.
                                                      \nc) If it is a factor, factorize\u00a0p(x)<\/em>\u00a0completely.
                                                      \nd) Find all the zeroes of\u00a0p(x)<\/em>.<\/p>\n

                                                      Prove your answer by providing a step-by-step solution and giving\u00a0three independent reasons<\/strong>\u00a0supporting your conclusion for part (b) from these domains:\u00a0(A) Direct Substitution, (B) Remainder Theorem Application, (C) Polynomial Long Division Verification.<\/strong><\/p>\n

                                                      [Solution: -]<\/strong><\/p>\n

                                                      a) Degree and Type<\/strong><\/p>\n

                                                        \n
                                                      • The highest power of x is 3. So, the\u00a0degree<\/strong>\u00a0is 3.<\/li>\n
                                                      • The polynomial has 4 terms:\u00a0\n
                                                        \"\"<\/p>\n

                                                        Source: Kapdec.com<\/p>\n<\/div>\n

                                                        . Therefore, it is simply called a\u00a0polynomial<\/strong>\u00a0(or specifically, a cubic polynomial).<\/li>\n<\/ul>\n

                                                        b) Verify if\u00a0<\/strong>(x-2)<\/em>\u00a0is a factor using the Factor Theorem.<\/strong><\/p>\n

                                                        The Factor Theorem states that\u00a0(x-a)<\/em>\u00a0is a factor of\u00a0p(x)<\/em>\u00a0if and only if\u00a0p(a)=0<\/em>. Here,\u00a0a=2<\/em>.<\/p>\n

                                                        (A) Direct Substitution<\/strong>
                                                        \nCompute\u00a0p(2)<\/em>:
                                                        \n=<\/em><\/p>\n

                                                        \"\"<\/p>\n

                                                        Source: Kapdec.com<\/p>\n<\/div>\n

                                                        <\/p>\n

                                                        \"\"<\/p>\n

                                                        Source: Kapdec.com<\/p>\n<\/div>\n

                                                        <\/p>\n

                                                        \"\"<\/p>\n

                                                        Source: Kapdec.com<\/p>\n<\/div>\n

                                                        <\/p>\n

                                                        \"\"<\/p>\n

                                                        Source: Kapdec.com<\/p>\n<\/div>\n

                                                        \"\"<\/p>\n

                                                        Source: Kapdec.com<\/p>\n<\/div>\n

                                                        Since\u00a0p(2)=-12<\/em>\u22600<\/em>, by the Factor Theorem,\u00a0(x-2)<\/em>\u00a0is\u00a0not<\/strong>\u00a0a factor.<\/p>\n

                                                        (B) Remainder Theorem Application<\/strong>
                                                        \nThe Remainder Theorem states that the remainder when\u00a0p(x)<\/em>\u00a0is divided by\u00a0(x-2)<\/em>\u00a0is\u00a0p(2)<\/em>. We calculated\u00a0p(2)=-12<\/em>. A non-zero remainder means that\u00a0(x-2)<\/em>\u00a0does not divide\u00a0p(x)<\/em>\u00a0exactly. Therefore, it is\u00a0not<\/strong>\u00a0a factor. This is a direct application of the theorem and is consistent with (A).<\/p>\n

                                                        \u00a0<\/p>\n

                                                        <\/p>\n