The topics covered here are errors that students often make
in doing algebra, and not just errors typically made in an algebra class. I’ve seen every one of these mistakes made by
students in all level of classes, from algebra classes up to senior level math
classes! In fact a few of the examples
in this section will actually come from calculus.
If you have not had
calculus you can ignore these examples.
In every case where I’ve given examples I’ve tried to include examples
from an algebra class as well as the occasion example from upper level courses
I’m convinced that many of these are mistakes given here are
caused by people getting lazy or getting in a hurry and not paying attention to
what they’re doing. By slowing down,
paying attention to what you’re doing and paying attention to proper notation
you can avoid the vast majority of these mistakes!
Division by Zero
Everyone knows that the problem is that far too many people also
say that or ! Remember that division by zero is
undefined! You simply cannot divide by
zero so don’t do it!
Here is a very good example of the kinds of havoc that can
arise when you divide by zero. See if you
can find the mistake that I made in the work below.
start assuming this to be true.
both sides by a.
- Subtract from both sides.
- Factor both sides.
both sides by .
we started off assuming .
both sides by b.
So, we’ve managed to prove that 1 = 2! Now, we know that’s not true so clearly we
made a mistake somewhere. Can you see
where the mistake was made?
The mistake was in step 5.
Recall that we started out with the assumption . However, if this is true then we have ! So, in step 5 we are really dividing by
That simple mistake led us to something that we knew wasn’t
true, however, in most cases your answer will not obviously be wrong. It will not always be clear that you are
dividing by zero, as was the case in this example. You need to be on the lookout for this kind
Remember that you CAN’T divide by zero!
This is probably error that I find to be the most
frustrating. There are a couple of
errors that people commonly make here.
The first error is that people get lazy and decide that
parenthesis aren’t needed at certain steps or that they can remember that the parenthesis
are supposed to be there. Of course the
problem here is that they often tend to forget about them in the very next
The other error is that students sometimes don’t understand
just how important parentheses really are.
This is often seen in errors made in exponentiation as my first couple
of examples show.
Example 1 Square
Note the very important difference between these two! When dealing with exponents remember that
only the quantity immediately to the left of the exponent gets the
exponent. So, in the incorrect case,
the x is the quantity immediately
to the left of the exponent so we are squaring only the x while the 4 isn’t squared.
In the correct case the parenthesis is immediately to the left of the
exponent so this signifies that everything inside the parenthesis should be
Parenthesis are required in this case to make sure we
square the whole thing, not just the x,
so don’t forget them!
Example 2 Square
This one is similar to the previous one, but has a
subtlety that causes problems on occasion.
Remember that only the quantity to the left of the exponent gets the
exponent. So, in the incorrect case
ONLY the 3 is to the left of the exponent and so ONLY the 3 gets squared!
Many people know that technically they are supposed to
square -3, but they get lazy and don’t write the parenthesis down on the
premise that they will remember them when the time comes to actually evaluate
it. However, it’s amazing how many of
these folks promptly forget about the parenthesis and write down -9 anyway!
Example 3 Subtract
Be careful and note the difference between the two! In the first case I put parenthesis around
then and in the second I didn’t. Since we are subtracting a polynomial we
need to make sure we subtract the WHOLE polynomial! The only way to make sure we do that
correctly is to put parenthesis around it.
Again, this is one of those errors that people do know
that technically the parenthesis should be there, but they don’t put them in
and promptly forget that they were there and do the subtraction incorrectly.
Example 4 Convert
to fractional exponents.
This comes back to same mistake in the first two. If only the quantity to the left of the
exponent gets the exponent. So, the
incorrect case is really and this is clearly NOT the original root.
Example 5 Evaluate
This is a calculus problem, so if you haven’t had calculus
you can ignore this example. However,
far too many of my calculus students make this mistake for me to ignore it.
Note the use of the parenthesis. The problem states that it is -3 times the
WHOLE integral not just the first term of the integral (as is done in the
Be careful when using the distribution property! There two main errors that I run across on a
Example 1 Multiply
Make sure that you distribute the 4 all the way through
the parenthesis! Too often people just
multiply the first term by the 4 and ignore the second term. This is especially true when the second
term is just a number. For some
reason, if the second term contains variables students will remember to do
the distribution correctly more often than not.
Example 2 Multiply
Remember that exponentiation must be performed BEFORE you
distribute any coefficients through the parenthesis!
I didn’t know what else to call this, but it’s an error that
many students make. Here’s the
assumption. Since then everything works like this. However, here is a whole list in which this
It’s not hard to convince yourself that any of these aren’t
true. Just pick a couple of numbers and
plug them in! For instance,
You will find the occasional set of numbers for which one of
these rules will work, but they don’t work for almost any randomly chosen pair
Note that there are far more examples where this additive assumption doesn’t work than
what I’ve listed here. I simply wrote
down the ones that I see most often.
Also a couple of those that I listed could be made more general. For instance,
These errors fall into two categories. Simplifying rational expressions and solving
equations. Let’s look at simplifying rational expressions first.
Example 1 Simplify
Notice that in order to cancel the x out of the denominator I first factored an
out of the numerator. You can only cancel something if it is
multiplied by the WHOLE numerator and denominator, or if IS the whole
numerator or denominator (as in the case of the denominator in our example).
Contrast this with the next example which contains a very
common error that students make.
Example 2 Simplify
Far too many
students try to simplify this as,
In other words,
they cancel the x in the
denominator against only one of the x’s
in the numerator (i.e. cancel the x only from the first term or only from the second term)..
THIS CAN’T BE DONE!!!!! In
order to do this canceling you MUST have an x in both terms.
To convince yourself that this kind of canceling isn’t
true consider the following number example.
Example 3 Simplify
This can easily be done just be doing the arithmetic as
However, let’s do an incorrect cancel similar to the
previous example. We’ll first cancel
the two in the denominator into the eight in the numerator. This is NOT CORRECT, but it mirrors the
canceling that was incorrectly done in the previous example. This gives,
Clearly these two aren’t the same! So you need to be careful with canceling!
Now, let’s take a quick look at canceling errors involved in
Example 4 Solve
Too many students get used to just canceling (i.e.
simplifying) things to make their life easier. So, the biggest mistake in solving this
kind of equation is to cancel an x
from both sides to get,
While, is a solution, there is another solution
that we’ve missed. Can you see what it
is? Take a look at the next example to
see what it is.
Example 5 Solve
Here’s the correct way to solve this equation. First get everything on one side then
From this we can see that either
In the second case we get the we got in the first attempt, but from the
first case we also get that we didn’t get in the first
attempt. Clearly will work in the equation and so is a
We missed the in the first attempt because we tried to make
our life easier by “simplifying” the equation before solving. While some simplification is a good and
necessary thing, you should NEVER divide out a term as we did in the first
attempt when solving. If you do this you
WILL loose solutions.
Proper Use of Square Root
There seems to be a very large misconception about the use
of square roots out there. Students seem
to be under the misconception that
This is not correct however.
Square roots are ALWAYS positive or zero! So the correct value is
This is the ONLY value of the square root! If we want the -4 then we do the following
Notice that I used parenthesis only to make the point on
just how the minus sign was appearing!
In general the middle two steps are omitted. So, if we want the negative value we have to
actually put in the minus sign!
I suppose that this misconception arises because they are
also asked to solve things like . Clearly the answer to this is and often they will solve by “taking the
square root” of both sides. There is a
missing step however. Here is the proper
solution technique for this problem.
Note that the shows up in the second step before we actually
find the value of the square root! It
doesn’t show up as part of taking the square root.
I feel that I need to point out that many instructors
(including myself on occasion) don’t help matters in that they will often omit
the second step and by doing so seem to imply that the is showing up because of the square root.
So remember that square roots ALWAYS return a positive
answer or zero. If you want a negative you’ll
need to put it in a minus sign BEFORE you take the square root.
This is more a notational issue than an algebra issue. I decided to put it here because too many
students come out of algebra classes without understanding this point. There are really three kinds of “bad”
notation that people often use with fractions that can lead to errors in work.
The first is using a “/” to denote a fraction, for instance
2/3. In this case there really isn’t a
problem with using a “/”, but what about 2/3x? This can be either of the two following
It is not clear from 2/3x
which of these two it should be! You, as
the student, may know which one of the two that you intended it to be, but a
grader won’t. Also, while you may know
which of the two you intended it to be when you wrote it down, will you still
know which of the two it is when you go back to look at the problem when you
You should only use a “/” for fractions when it will be
clear and obvious to everyone, not just you, how the fraction should be
The next notational problem I see fairly regularly is people
writing . It is not clear from this if the x belongs in the denominator or the
fraction or not. Students often write
fractions like this and usually they mean that the x shouldn’t be in the denominator.
The problem is on a quick glance it often looks like it should be in the
denominator and the student just didn’t draw the fraction bar over far enough.
If you intend for the x
to be in the denominator then write it as such that way, ,
i.e. make sure that you draw the
fraction bar over the WHOLE denominator.
If you don’t intend for it to be in the denominator then don’t leave any
doubt! Write it as .
The final notational problem that I see comes back to using
a “/” to denote a fraction, but is really a parenthesis problem. This involves fractions like
Often students who use “/” to denote fractions will write
this is fraction as
These students know that they are writing down the original
fraction. However, almost anyone else
will see the following
This is definitely NOT the original fraction. So, if you MUST use “/” to denote fractions
use parenthesis to make it clear what is the numerator and what is the
denominator. So, you should write it as