Question

Create a quadratic polynomial function f(x) and a linear binomial in the form (x − a).

Answer 1

Quadratic polynomial  x^2 - x - 6 = f(x)
binomial  (x-3)

dividing:-
                x   + 2
          ----------------------
x - 3   )  x^2  - x   -  6
             x^2 - 3x
             ----------
                     2x - 6
                     2x - 6
                     --------
                      .........

result of division is (x + 2)  
s o we can write the function as (x - 3)(x + 2)

Part 2  f(a)  = f(3)   =  (3)^2 - 3 - 6 = 0

Part3  The remainder Theorem  states that  the remainder when a polynomial is divided by (x - a)  then the remainder is f(a).  If f(a) = 0 then  x - a is a factor.
In the above case f(3) = 0 therefore (x - 3) is a factor of our polynomial.