Polynomials-types, Graphs, Factor and Remainder theorem

Polynomials-types, Graphs, Factor and Remainder theorem

H.C.F and L.C.M of Polynomials :
If a polynomial p(x) is a product of two polynomials h(x) and g(x)i.e.,
\( f(x) = g(x) \times h(x) \) then h(x) and g(x) and are said to be factor of f(x).
Ex : Let  \( f(x) = x^2 - 7x + 10 \) 
  \( f(x) = (x - 2)(x - 5) \)     
Note : If h(x) is a factor of f(x)
       \( \therefore \) -h(x) is a factor of f(x).

Highest common factor (H,C,F) or Greatest common divisior (G.C.D):
The product of the least powers of the common factors is said to be H.C.F of the given polynomials.
Let \( f(x) = (x - 1)^2 (x - 2).(x - 3)^3 = (x - 1)^2 .(x - 2)^1 .(x - 3)^3 (x - 4)^0 \)   
     \( g(x) = (x - 1)^3 (x - 4) = (x - 1)^3 .(x - 2)^0 .(x - 3)^0 .(x - 4)^1 \)   
     \(\therefore\)H.C.F = \( (x - 1)^2 .(x - 2)^0 .(x - 3)^0 .(x - 4)^0 = (x - 1)^2 \times 1 \times 1 \times 1 = (x - 1)^2 \)
Least common multiple of polynomials (L.C.M)
    The product of the highest powers of common factors is said to be L.C.M of the given polynomials.
    Let  \( f(x) = (x - 1)^1 .(x - 2)^2 .(x - 4)^1 \)  
    \( g(x) = (x - 1)^3 .(x - 2)^1 .(x - 4)^3 \)
    \( \therefore \,\,\,\,L.C.M = (x - 1)^3 .(x - 2)^2 .(x - 4)^3 \)