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Chapter 3: Polynomials

The Basics  

    Polynomial is an equation that is formed by adding or subtracting several variables called monomial. Monomial is a variable that is formed with a number and a letter variable to its powers. The example of monomial is 3X³. You can’t add or subtract monomials if they have different exponents such as 3X³ and 4X⁴. But you can multiply or divide them. To multiply monomials, just add the exponents of the variables and multiply the coefficients. 3X³ x 4X⁴ = 12X⁷.

  Here are some additional ways to manipulate the monomials:

•         (a^m)ⁿ = a^mn
•         (ab)^m= a^mb^m
•  

  Expansion

      Most time when you see a polynomial, it is expanded. But what about its non-expanded forms. The most commonly seen two types of non-expanded polynomials are (ax+b)(cx+d) and (a+b)^c. When expand the first type, you just multiply it out. The second type is actually a special case of the first type with same terms inside each bracket. To expand the second type, you can just multiply it out. But it takes time.

      There is an easy way to do this, and it takes a little memorization. The easy way to do this is called the Pascal’s Triangle. The Pascal’s Triangle is a series of numbers that are putted in the form of a triangle. The first row has one number; the second row has two numbers; the third row has three numbers and so on. Each number in the triangle is the sum of the two numbers above it. By memorizing the Pascal’s Triangle, you can solve the most common polynomial expansions easily and accurately. For example: when you want to expand the polynomial (a+b)³, the coefficient of each expanded term is actually the number on the forth row of the Pascal’s Triangle. (a+b)³ = a³ + 3a²b + 3ab² + b³.  So we can conclude that the coefficients of the expanded polynomials are the numbers of corresponding row minus 1.

  Pascal's Triangle

  1
  1        1
  1          2           1
  1            3          3             1
    1          4             6            4            1
  1          5          10         10              5       1
         1          6        15              20             15         6         1    

  .....

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