Whether it is a calculus class or a first algebra class, I have seen students at every level and every natural ability make these mistakes. Why would so many fall into these traps? Well, it really comes down to working too quickly and applying rules without stopping to think about whether or not they apply. If you can avoid these mistakes, you will find yourself well ahead of the curve. In fact, these are the kinds of things people writing multiple choice tests think about.
 Distributing a Power
It seems as though everyone wants , and who wouldn’t? Every math exercise in the world becomes much easier if you use this! Unfortunately, in general, . Its true in a trivial case – when a or b equals zero (plug zero in for either one to check). Otherwise, this is a no go and if you aren’t sure, try it out with some nonzero numbers: What if a=1 and b=1?while . Clearly we get two different things. Now this isn’t a mathematical proof since I used two specific numbers but the fact that it does not work here should give you pause. In fact, you would have to FOIL the left hand side to find its equivalent expression. In other words, .
 Forgetting to Completely Distribute
What if you needed to simplify . Many students would write . Can you see what is wrong?The negative in front of the parentheses is basically a negative 1 multiplying all of the terms within those parentheses. That means it must be distributed to EVERYTHING, not just the first term! . This works regardless of what is in front of the parentheses. For instance, .
Most often, it seems that people understand this but make this mistake when they are going too fast. This is why I advocate writing down every single step when you do any math problem. (or just double check yourself anytime you have had to distribute)

Applying the ZeroProduct Rule to Everything
The zero product rule is what allows you to solve an equation like . By realizing that the only way two things can multiply to give us zero is if one or both are zero, we can say and . This is all based on the fact that there is a zero on the right hand side!If instead, we had we couldn’t immediately take this approach. This does not imply that and . Again, you can only apply this if you manage to get a product of terms equal to zero. (to solve the new equation, you could foil the left hand side and then bring the two over – going from there)
There are plenty of other mistakes that I have seen but these are by far the most common. In the end, it comes down to applying properties in situations where they do not apply. Learn to ask yourself if the rule you are applying makes sense in a given situation and you will be well on your way to avoiding these!
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