Question

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.

Part 2. Show all work to evaluate f(a) using the function you created.
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

Answer 1

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 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.