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POLYNOMIALS - Coggle Diagram

- - - - P(x) is the dividend polynomial.
        
        Q(x) is the divisor polynomial.
        
        B(x) is the quotient polynomial.
        
        K(x) is the remainder polynomial.
      - According to the division algorithm above, we get the following:
        
        P(x) = Q(x).B(x) + K(x)
        
        If K(x) = 0, it is said that P(x) is divided by Q(x) without remainder.
        
        deg[K(x)] = 0 < deg[Q(x)]
        
        deg[B(x)] = deg[P(x)] - deg[Q(x)]
    - - Arrange both the dividend polynomial and the divisor polynomial in descending powers of the variable.
      - The term with the highest degree of dividend polynomial is divided by the term with the highest degree of divisor polynomial and the result is written as a first term of quotient polynomial.
      - The first term of quotient polynomial is multiplied with divisor polynomial and the result is subtracted from dividend polynomial.
      - All these operations are done until the degree of remainder polynomial is smaller than the degree of divisor polynomial.
- - - - The first step in completely factoring a polynomial is to remove (factor out) any common factors.
      - For the polynomials A(x), B(x), and C(x), A(x).B(x) + A(x).C(x) = A(x).[B(x)+C(x)]
    - - If there is no common number, common variable or common term in each term of a given polynomial, the terms that have any common factors are made a group by coming together. Then, each group are factored out such that they have the same expression in parentheses. After that, the groups are factored out any common factor.
      - ax + ay + bx + by = a(x+y) + b(x+y) = (a+b).(x+y)
      - ax + ay + bx = (a+b)x + (a+b)y = (a+b).(x+y)
    - - a) Perfect Square
        
        The expression thats in the form of (x+y)2 or (x-y)2 are called perfect square expressions.
        
        (x+y)2 = (x+y).(x+y) = x2+xy+yx+y2 = x2+xy+y2
        
        (x-y)2 = (x-y).(x-y) = x2-xy-yx+y2 = x2-2xy+y2
      - b) Difference of Two Squares
        
        The difference of two squares is the form of x2-y2.
        
        x2-y2 = (x-y).(x+y)
      - c) Factoring the Cube of Sum and Difference
        
        (x+y)3 = (x+y).(x+y).(x+y) = x3+3x2y+y3. Then we also get x3+y3 = (x+y)-3x2y-3xy2 = (x+y)3-3xy(x+y)
        
        (x-y)3 = (x-y).(x-y).(x-y) = x3-3x2y+3xy2-y3. Then we also get x3-y3 = (x-y)3+3x2y-3xy2 = (x-y)3+3xy(x-y)
      - d) Factoring the Sum and the Difference of Two Cubes
        
        x3+y3=(x+y).(x2-xy+y2)
        
        x3-y3 = (x-y).(x2+xy+y2)
    - - For ≠ 0 and a,b,c ∈ R, by factoring three term polynomials of the form ax2+bx+c, we look at the factors of a and c.
      - For a=k.t and c=m.n,
        
        If kn+tm = b, it is said that ax2+bx+c = (kx+m).(tx+n)
    - - It is the process of simplifying the polynomial by renaming similar terms with a new variable.
- - - - a) Addition Operation
      - b) Subtraction Operation