Farewell, Synthetic Division

I am so happy to see synthetic division fade off the Algebra II planet as high schools morph slowly into extensible mathematics.  Synthetic division got its popularity by being faster, easier, and more entertaining than long division with polynomials.  It is disappearing because the Common Core writers had the nerve to call it out for what it is:  a huge distraction from learning  the kinds of mathematics that transfer to other contexts.

For one thing, synthetic division only works with certain kinds of polynomial division. Students still need long division for other cases.  What happens, though, is most students hear synthetic division – long division as one is easier than the other:  so they  earn the points for the easy route (synthetic) and don’t worry about a few points lost for not bothering  with the more challenging (long division).  Students who try to do both often find themselves doubly confused because with synthetic division, we add; and with long division, we subtract…and subtract negatives.

There is a ton of useful math to be reviewed within the process of long division:

  • add/subtract like terms
  • when subtracting with a negative, that’s the opposite of subtracting (add)
  • when the divisor is a sum, the quotient is distributed onto the divisor to get the dividend
  • long division can be used to factor the sum or difference of cubes (the divisor is found via the same logic used to factor a difference of squares).

The argument for keeping synthetic division around has typically been one of “speed for finding roots.” The same educators are often in favor of memorizing formulas for factoring cubes.   I would argue that if speed is needed for finding roots, then it is far more efficient to look at the graph and firmly establish those relationships.

There are benefits from being able to program (code) registers and matrix operations such as synthetic division.  It makes good sense to teach that in a linear algebra course beyond math analysis or pre-calculus.  In Linear Algebra, one might expect to connect synthetic division with other matrix operations.  In the mean time, I argue the focus in high school Algebra II needs to be on long division as applied to polynomials and factoring cubes.  For those who would maintain allegiance to math speed and synthetic division, I would ask how much time they devote to modeling and problem-based learning with the class-time saved….let me guess.  In the mean time, I recommend  Nixthetricks.com . That resource nixed a few others of mine.


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