Name:
Dynamics of Algebra II
Directions: Use synthetic substitution to find and for each polynomial.

1. 
   a.
   
   
   

To evaluate using synthetic substitution, write down the coefficients of the polynomial: 1, 2, -3, and 1. Then, set up synthetic substitution:

-5 |  1   2  -3   1
   |         -5   15  -10
-------------------------
   |  1  -3  12  -9

The final value in the bottom row represents , which is -9.

Now, to find , do the same with 2:

2 |  1   2  -3   1
   |         2   8   10
-------------------------
   |  1   4   5  11

So, .

Now you’ve found and !

Understanding synthetic substitution can be tied to its historical roots in polynomial long division. Originating in the work of mathematicians like Horner, this method helps simplify calculations, allowing for quicker evaluations of polynomials. It’s essentially a clever way to avoid the more tedious process of long division!

When it comes to real-world applications, synthetic substitution is invaluable for computer algorithms that deal with polynomial evaluations in fields like graphics programming, physics simulations, and even financial modeling. It allows for quick calculations that can make extensive computations more manageable and efficient. Being able to utilize this technique can really streamline problem-solving in practical scenarios.