Pre-Calculus For Dummies, 2nd Edition (2012)
Part IV. The Part of Tens
Chapter 17. Ten Habits to Quit Before Calculus
In This Chapter
Recognizing all the bad habits
Remembering all the correct habits
You’ve come this far in math, for which you deserve high praise. But we can say with confidence that only 1 percent of you have few to no bad math habits. We wrote this chapter on breaking bad math habits in hopes that you’ll recognize a mistake that you commonly make and resolve to not make it anymore. The 1 percent of you who are perfect can now leave the building!
Operating Out of Order
Don’t fall for the trap that many students do by performing operations in order from left to right. For instance, 2 – 6 · 3 doesn’t become –4 · 3, or –12. You’ll reach those incorrect answers if you forget to do the multiplication first. Focus on following the order of PEMDAS every time, all the time:
Parentheses (and other grouping devices)
Multiplication and Division from left to right
Addition and Subtraction from left to right
Don’t ever go out of order, and that’s an order!
Squaring without FOILing
Always remember to FOIL. Here’s a common mistake we see: (x + 3)2 = x2 + 9. Why is that wrong? Because you forgot that to square something means to multiply it times itself. Foiled again! Here’s the correct process: (x + 3)2= (x + 3)(x + 3). You should reach the answer of x2 + 6x + 9 for this one. Don’t be a square and forget to FOIL. (See Chapter 4 for a review of the FOIL method.)
Splitting Up Terms in Denominators
Working with denominators can be tricky. We’re here to tell you that
Look at this situation with numbers only; say x = 1, like this expression:
If you split up the denominator, you get , which should be .
But notice that you forgot to do PEMDAS (see “Operating Out of Order” earlier in this chapter). That division bar is a grouping symbol. You have to simplify what’s on the top and what’s on the bottom before doing the division.
The answer should be , or 1. The same warning applies to variables, too,
like in .
Because variables represent numbers (albeit undetermined ones), they have to follow the same rules as numbers do.
Combining the Wrong Terms
Recognizing like terms can be fairly simple or annoyingly difficult. On the simple side, you must realize that you can’t add a variable and a constant — 4x + 3 isn’t 7x, no matter how hard you try. These terms are not like. (That’s like saying that four chickens plus three dogs is seven chogs. Not quite!)
The more complicated the polynomial, the more likely you are to combine unlike terms, so be especially careful when dealing with doozies. For example, 4a2b3 – a3b2 is in simplified form. Even though the two terms look remarkably similar, they’re not like terms because they don’t have all of the same variables with the same exponents, so you can’t combine them. The expression 4a2b3 – a2b3, however, is a different story. This expression has like terms, and you can find their difference: 3a2b3.
Forgetting the Reciprocal
The chances of making mistakes increase when you’re working with complex fractions — particularly when it comes to dividing them. You’re moving right along, and then this expression:
suddenly becomes this:
You forgot to do the division! The big bar between the two fractions means “divide!”, so you really should do this:
Dividing a fraction means to multiply by its reciprocal:
When this expression simplifies, you get
Losing Track of Minus Signs
When subtracting polynomials, most people tend to lose minus signs. For example, when you work on this problem
(4x2 – 6x + 3) – (2x2 – 4x + 3)
you may have a tendency to write
4x2 – 6x + 3 – 2x2 – 4x + 3, or 2x2 – 10x + 6
But you forgot to subtract the whole second polynomial; you only subtracted the first term. Make sure that the minus sign in front of the second polynomial comes out to play in each subsequent term. The right way to work this problem is 4x2 – 6x + 3 – 2x2 + 4x – 3, which becomes 2x2 – 2x.
One common mistake like this occurs when subtracting rational functions, as in this example:
Many people turn this expression into
but if they do, they forget to subtract the whole second polynomial in the numerator. The problem should become
which eventually simplifies to
People tend to oversimplify radicals when progressing through the stages of math. Some common mistakes include losing the root sign altogether, so that becomes just 3. Some people even toss the index so that becomes , which is 2 . . . except that it’s not. The cube root of 4 is approximately 1.5874.
Other mistakes include adding roots that shouldn’t be added — like , or . These two roots aren’t like terms, so you can’t add them (see the earlier section “Combining the Wrong Terms”).
When working with radicals, be sure to always simplify, but make sure you don’t fall for the common traps described here.
Erring in Exponential Dealings
Multiplying monomials with exponents doesn’t mean you multiply their exponents — you add them instead. For instance, x3 · x4 isn’t x12; it’s x7. Similarly, when finding the power of a product, don’t forget to apply the power to every term by multiplying the exponents. For example, (3x4y2)2 isn’t 3x8y4, because you forgot the power on the 3. It should be 9x8y4.
Watch for negatives — especially on calculators. The monomials –32 and (–3)2 represent –9 and 9, respectively, because the order of operations says that you must take an exponent first.
Canceling Out Too Quickly
People cancel incorrectly in so many different ways, we could write a whole For Dummies book to cover them all. Instead, we just look at the most common mistakes in the following list:
When dealing with rational expressions, you can’t simplify constants by throwing the distributive property out the window. When canceling terms on the numerator and denominator, the bottom term must go into every term on the top. Division is a lot like multiplication; if you have an expression with more than one term in the numerator, you must make sure that the term on the bottom divides evenly into all terms on the top.
For example, consider this expression:
It doesn’t equal 2x – 1, because the 2 on the bottom doesn’t divide evenly with both 4x and –1. Therefore, this rational expression doesn’t simplify; you can leave it alone. If you’re going to divide the 2, the result had better be 2x – 1/2.
You can’t get cancel happy and cross out variables from terms.
Take a look at this expression:
It doesn’t simplify to 3x – 6 + 1, or 3x – 5, by canceling the x terms and the 2s. Instead, you can rewrite the fraction as
which simplifies to
Don’t cancel multiple terms that can’t be cancelled on the top and bottom of a rational expression. After you’ve factored any factorable polynomials and cancelled like terms, you’re done. For example, we’ve seen students cancel expressions like
by doing this:
You have to factor first to get
which reduces by canceling (x – 3) to obtain
Now you’re done. Please don’t fall for the trap and simplify it any further.
Even if a rational expression has no factoring to do, don’t cancel if you can’t. For instance, this expression has no factoring to do:
You can’t start canceling like this:
Remembering that variables represent numbers, you have to follow the order of operations to fully simplify the numerator and the denominator before you can divide (or cancel). You can’t simplify on the top or the bottom of the term in any way, so you can’t do any canceling.
When distributing, don’t forget to multiply the term you’re distributing to every single term within the parentheses. (Think of how the negative sign in a subtraction applies to every term.) For instance, 2(4x2 – 3x + 1) doesn’t equal 8x2 – 3x + 1 or even 8x2 – 6x + 1; it equals 8x2 – 6x + 2.