Iwill give 50 points and brainliest answer if you just ! its only 8th grade so it shouldnt be that ! !
task 5—polynomial division and the remainder theorem
create a quadratic polynomial function f(x) and a linear binomial in the form (x − a).
part 1. show all work using long division to divide your polynomial by the binomial.
part 2. show all work to evaluate f(a) using the function you created.
part 3. use complete sentences to explain how the remainder theorem is used to determine whether your linear binomial is a factor of your polynomial function

Suppose the quadratic equation is x²+2x+1 and the binomial is x+1.

Part 1. In executing the long division method, you merely use the form like what were taught to in elementary:

                                 Quotient
                               
                      divisor| DIvidend

But in this case, we deal with polynomials instead of only numbers. The complete solution is

                                 x + 1
                             
                       x+1 | x²+2x+1 
                              -x²+x
                               
                                0 + x+1
                               -      x+1
                               
                                         0
So, the answer is  x+1.

Part 2. The quadratic equation is f(a) = x²+2x+1, where a is the constant in the binomial x-1. Thus, a=1. Substituting a to the x values, f(1) = (1)²+2(1)+1=0. The zero means that there is no remainder. Thus, (x+1) is a factor.

Part 3. The remainder theorem works similar to Part 2. If you substitute f(a) and the answer is a non-zero number, then that is the remainder when you divide the quadratic equation with the binomial. Only when the answer is zero that you can confirm that the linear binomial is a factor.