Synthetic Division and the Remainder Theorem

Hey, it's Breanne. If you don't know who I am I'm that girl that sits at the back of the classroom in the corner, eating. 

Synthetic Division 
So to refresh your memory synthetic division is a method of performing polynomial division, with less writing and fewer calculations.

Synthetic division is actually super easy, but you really need to master these key steps in order to do it correctly. 

1. Arrange the coefficients in descending order according to the powers of x (if there isn't a power, write 0) 
2. After writing the divisor in the form x - a, use "a" to generate the second and third rows of numbers as follows. Bring down the first coefficient of the dividend.  Multiply this sum by "a", and add the product to the third coefficient of the dividend. Repeat the process until a product is added to the constant term of f(x).
3. The last number in the third row of numbers is the remainder; the other numbers in the third row are the coefficients of the quotient, which is of degree 1 less than f(x).

So in english, you get this equation:

You subtract a from x to get the divisor.

You then write the coefficients down and bring down the first number. 

Multiply the divisor by the first number you brought down. When this is done you carry the result to the next column of numbers. 

You then add down the row.

You continue these steps until you've reached the last number.

You write it out and then simplify, and you're done! 

The Remainder Theorem 

So during our last class we learned about the remainder theorem. We started off doing examples using synthetic division, but always ending up with a remainder. We would then plug "a" into the equation and we get the remainder. 

   

After doing a couple of examples we came up with this conclusion:

When a polynomial P(x), is divided by any binomial, (x-a) we can write the statement: P(a) = R

I hope his helped everyone and I hope your thanksgiving weekend was wonderful!!! 

-Breanne Calvadores