Create a quadratic polynomial function f(x) and a linear binomial in the form (x − a).
x^2 – 3x + 6 and x-9
part 1. show all work using long division to divide your polynomial by the binomial.
x – 9 = 0
x = 9
first take the original expression.
x^2 - 3x + 6
fill in the blanks.
9^2-3(9)+6
81-27+6
81-21
60
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
the remainder theorem says when a polynomial is being divided by a linear binomial, the remainder is equal to f(a). the remainder must be equal to zero. but the remainder is sixty, and not zero, so the linear binomial, x-9, would not be a factor of x^2-3x+6.

Part 1. In the long division, you find the greatest factor that could divide the dividend. You do this one at a time per term. Then, you find the product of the factor and the divisor, then subtract it from the dividend. The cycle goes on until all the terms are divided:
               x  +  6
           
    x - 9 |  x² - 3x + 6
             - x² - 9x
             
                       6x + 6
                     - 6x - 54
                     
                           60

There quotient is (x+6) with a quotient of 60.

Part 2.  The solution is already given. Set the binomial to 0, such that x-a = 0 is equal to x=a. Using this a to substitute the x terms in the given function:
f(9) = (9)² - 3(9) + 6
f(9) = 60

Part 3. The steps shown are from the concept of Factor and Remainder Theorem. When you substitute x=a to the function, the answer could determine if x=a is a factor or not. If the answer is zero, then x=a is a factor. If not, the answer represents the remainder.

Therefore, x = 9 is not a factor of the given function. It yields a remainder of 60 which coincides with Part 1.