Question

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

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 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.

Answer 2

Answer:

Step-by-step explanation:

Part 1. Show all work using long division to divide your polynomial by the binomial.

                      X + 8

          ----------------------------

X-10 |   x^2 - 4x + 8

           (-x^2 + 10x)

             6x + 8

             (-6x + 8)

              68

X + 8 + 68 / x - 10

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

f(x) = x^2 - 4x + 8

f(10) = (10)^2 - 4(10) + 8

f(10) = 68

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.

When you substitute f(x) to f(a) you would be able to use this to determine if f(x) to f(a) is a factor or not. If the answer is zero then it is not a factor, if the answer is not zero then it will have a remainder.