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### Course: Algebra (all content)>Unit 10

Lesson 22: Synthetic division of polynomials

# Intro to polynomial synthetic division

Sal shows how synthetic division of polynomials works. Created by Sal Khan.

## Want to join the conversation?

• Which method is better in dividing polynomials? Synthetic or Long Division.
• Personal I love Synthetic division, try practicing with for a few days to a week until you understand how it really works because on a test or exam is so fast to do polynomial division with Synthetic division. Only downside to Synthetic division is that sometimes people have no idea what's going on it's almost like voodoo.
• is it necessary to know synthetic division for exams? (I do the IB)
• You don't really need to know synthetic division, or if you do, you should know long division too. It can save you valuable minutes, though.
Synthetic division reminds me of a FOIL - a thing from algebra to remember on how to distribute something like: (x+4)(x-3). Problem with Synthetic division and FOIL, is that they work only in couple simple cases and not in complex situations.

Edit: Apparently, I was wrong to some extent. Synthetic division proves to be useful when factoring polynomials what have more than two roots, e.g. x^4+2x^3+x-1=0. I won't go into a detail, but in terms of speed when you need to check like 6 roots, you can easily check them in half the time, compared to a long division.
• but what if youre deciding if something like 2x-1 is a factor of a polynomial function? i know you have to make it into the (x-k) form but how??
• You divide out the coefficient of x, to get a divisor of the form (x-k); you can then use synthetic division to check if (x-k) is a factor of the polynomial.
Here 2x-1 = 2(x-1/2), so you can use synthetic division to divide (x-1/2) into the polynomial to see if it is a factor, which would be equivalent to (2x-1) being a factor.
• Is this the same as Synthetic Substitution? because my teacher is calling it that. It seems to be the same thing.
• What if the DENOMINATOR is not in 1st degree, but in 4th or 6th degree?
• My math teacher told me that this method can only be used for a linear denominator, such as x-k. So you can't use this for anything but a first degree denominator
• At in this video I think I see a problem. Sal says that you multiply the 30 times a negative four when there is no negative four on the screen! Is this just me? Or if not, how did Sal get the negative four?
(1 vote)
• What is the difference between synthetic division and just division of polynomials?
• synthetic division is a big short cut if the divisor is only an X +/- any number.
(1 vote)
• can you not do synthetic division when the x is squared? how do you divide two functions when it is squared?
• Use polynomial long division if the divisor has x^2 or higher term.
• Can we use synthetic division to solve polynomial that has more than one variable in numerator?
example: 2x^3 + cx^2 + 2x + 2 divided by x-2
• this is if you have 1 variable
you need the remainder, the last value in the synthetic division ( for the question in the video it was -121) and then plug it in the original equation with x=2 and y= the remainder.
You could work backward using synthetic division but its harder especially if you are new at this but to do that way you also need the remainder
if you have two variable you need 1 extra x and y value
if you have three variable you need 2 extra x and y value (3 x and y value in total)
and so on