Modules for Prerequisite Mathematics

Subsection 2.1.1 Factoring Numbers

Once we are comfortable with variables, there are a variety of useful techniques, tools and tricks for manipulating expressions involving constants and variables. In this module, we'll go over a selection these tools and tricks. The first is various forms of factoring.

Factoring comes from the properties of multiplication of numbers. Consider this simple addition.

All of the terms are divisible by 3, so we can write them as multiples of 3.

Factoring is pulling the 3 out of the entire expression (using the distributive law).

Thus we have isolated the common factor, 3, from the left side of the equation. We could then divide both sides by this isolated 3.

Subsection 2.1.2 Factoring with Variables

Factoring is not terribly intense with numbers. We are more interesting in factoring with variables. However, when we factor with variables, we are not doing anything essentially different from what we just did here with numbers.

There are a vareity of situations where we might want to factor out a variable. We might be solving an equation where we can remove a variable.

Here, we can factor the variable \(x\) out of the left side of the equation.

If we used the distributive law to distribute the factored \(x\) again, we would recover the original expression. (If you want to check if you've done your factoring correctly, multiply the factor back in and see that it matched the original expression.) After factoring, we can divide by the variable \(x\) to lead to an easier equation.

This gives us the solution \(x=14\text{.}\) However, we also have the solution \(x=0\text{.}\) It is important to remember anytime you divide by a variable, or cancel off a variables (which is the same thing), you have to consider the alternatively possibility that the variable might be zero. Our equation here has two solutions: \(x = 14\) and \(x = 0\text{.}\)

Sometimes we might want to factor something out of the numerator and denonminator of a fraction to try to simplify the fraction. Here is an example, where \(p\) is the variable.

The variable is a common factor to both numerator and denominator, so we can factor it out of both.

Then, since \(p\) is a factor of the whole numbers and of the whole denominator, we can cancel it out.

The reason why we can do this is that any expression \(\frac{p}{p}\) evaulates to 1 (assuming that \(p \neq 0\)). Multiplying by 1 does do anything, so the fraction with and the fraction with \(\frac{p}{p}\) is the same thing.

Subsection 2.1.3 More Complicated Factors

We can do more than just factor out a variable. Sometimes we can factor out a binomial (such as \((x-9)\)), or some other more complicated expression. The more complicated the term, the more difficult it can be to recognize the factor. There are no simply, algorithmic rules for this kind of factors. Much of the skill comes from practice and familiarity with a variety of mathematical situations. I'll give a couple examples here, but don't assume that all problems will look like these examples; they are meant to demonstrate but not be representative.

First let's look at an expression where we can factor out a binomial.

Here there is a pattern with repeated 4s and 2s. I'm going to try to produce a common binomial in the first two and second two terms by factoring.

Applying the distributive law to both of these factors would produce the original expression (in the second, multiplying by 1 has no effect). Then, if we think of the term (4x-2) as a thing, we have \(x\) times that thing plus 1 times that thing. Therefore, we can write this expression as

This is a factorization of the original expression. Both the binomials \((x+1)\) and \((4x-2)\) are factors of the expression. This is an example of factoring a quadratic, which is discussed in more detail in Chapter 13. For now, let me make one more point on this example. Say we had this quadratic.

Then, in the middle, I could add \(2x\) and subtract the same, which doesn't change anything. (I can add and subract any quantity I want; adding and subtracting the same thing has no effect on any expression.)

Now I can group the third and fourth terms.

Now I have the same expression that I started with, which factors as above. Therefore, I can conclude that

The point of this remark is that often various ways to factor an expression are hidden. Helpful factorizations are often not obvious and can, in fact, be very difficult to find. As I said before, this isn't algorithmic. Sometimes, creativity and persistence is required to find the right way to manipulate an expressions.

Subsection 2.1.4 Factoring with Roots

Sometimes we need want to factor things out of a square root (or other root). Let's start with numbers and consider this square root.

We can write 40 as \((4)(10)\text{.}\)

Then it is a property of square root (since they are essentially exponents, using the properties of exponents in Chapter 16) that we can split up a square root over multiplication or division.

Then we can evaluate the first part: the square root of 4 is 2.

What we have done here is essentially factored out 4 from the square root. But when we took it out of the square root, we had to apply the square root, so 4 became 2. This is a useful technique for writing square roots in simply form, since any factor which has a nice square root can be removed. Here are a couple more examples.

We can do the same thing with variables and roots. Here is an expression with variables inside a square root.

We can factor the term inside the square root.

Then we can split up the square root over the multiplication.

Then we can get rid of the first square root, as we did with numbers above. However, there is a subltety here with variables. We can't just write \(\sqrt{x^2} = x\text{.}\) If the variable turned out to have a negative value, say -3, then we would square it to produce 9 and square root to produce 3. We would have lost the negative sign. To recognize this, we use the absolute value of the variable (discusssed in more detail in Section 4.1).

We've factored the \(x^2\) out of the square root; in the process, by applying the square root, it became \(|x|\text{.}\)

Subsection 2.1.5 Some Distribution Cautions

The distributive law of multiplication/division over subtraction/addition is fundamental. In my experience, it is intuitively well understood by most students. In this section, I want to talk about the danger of that intuitive understanding. Because distribution seems so natural and reasonable, students will often try to distribute where it isn't appropriate. This section is a series of cautions about overuse of the distributive law.

First to recap. Distribution is necessay for multiplication and division.

If we do the addition give, we get \(9 + 15 = 24\text{,}\) and then dividing by 4 gives 6. If we distribute, we get a sum of fraction (discussed in Chapter 5). The sum of fraction, using common denominator, gives \(\frac{24}{6}\text{,}\) which is again 6. The distributive law works here.

Consider these next two expression.

This looks like distribution: we've distributed the square root over the sum of the two numbers. However, let's check the values. The left side, if we do the sum, is \(\sqrt{25} = 5\text{.}\) The right side, when we do the square roots and then the sum, is \(4 + 3 = 7\text{.}\) 5 is not 7, so these are not equal.

I could restate this with variables.

With variable, this might be less obvious. That brings me to an important point.

IF YOU WANT TO CHECK IS AN ALGEBRAIC MANIPULATION WORKS, TRY INSERTING NUMBERS FOR THE VARIABLES AND SEE IF IT HOLDS.

Square roots are not the only place where students often use distribution incorrectly. Here are some other common places where distribution fails. For all of these, we show the problem by choosing numbers for the variables and explicitly calculating the right and left sides. We will get different values, so the two sides cannot be equal.

• Distribution does not work for denominators, as mentioned already in Section 1.1.
  \begin{equation*} \frac{1}{a + b} \neq \frac{1}{a} + \frac{1}{b} \end{equation*}
  We can check this by using \(a = 2\) and \(b = 3\text{.}\) The left side evalutes to
  \begin{equation*} \frac{1}{2 + 3} = \frac{1}{5} \text{.} \end{equation*}
  The right side evalutes to
  \begin{equation*} \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \text{.} \end{equation*}
  \(\frac{1}{5}\) is not the same as \(\frac{5}{6}\text{.}\)
• Distribution does not work for exponents (covered in Chapter 16).
  \begin{equation*} (a + b)^2 \neq a^2 + b^2 \end{equation*}
  We can check this by using \(a = 2\) and \(b = 3\text{.}\) The left side evalutes to
  \begin{equation*} (2 + 3)^2 = 5^2 = 25\text{.} \end{equation*}
  The right side evalutes to
  \begin{equation*} 2^2 + 3^2 = 4 + 9 = 13 \text{.} \end{equation*}
  25 is not the same thing as 13.
• Distribution does not work for logarithms (covered in Chapter 18).
  \begin{equation*} \log_2 (a + b) \neq \log_2 a + \log_2 b \end{equation*}
  We can check this by using \(a = 2\) and \(b = 4\text{.}\) This is a little bit less obvious, since we have ask a calcluate or computer for the value of the logarithm on the left side.
  \begin{equation*} \log_2 (2 + 4) = \log_2 6 \doteq 2.585\text{.} \end{equation*}
  The right side can be exactly evaluated.
  \begin{equation*} \log_2 2 + \log_2 4 = 1 + 2 = 3 \text{.} \end{equation*}
  The approximate value 2.585 is certainly not 3.
• Distribution does not work for trigonometric functions (covered in Chapter 15).
  \begin{equation*} \sin (a + b) \neq \sin a + \sin b \end{equation*}
  We can check this by using \(a = \frac{\pi}{2}\) and \(b = \frac{\pi}{2}\text{.}\) The left side evalutes to
  \begin{equation*} \sin \left( \frac{\pi}{2} + \frac{\pi}{2} \right) = \sin \pi = 0 \text{.} \end{equation*}
  The right side evalutes to
  \begin{equation*} \sin \left( \frac{\pi}{2} \right) + \sin \left( \frac{\pi}{2} \right) = 1 + 1 = 2 \text{.} \end{equation*}
  0 is not the same as 2.

Subsection 2.1.6 Simplification

This factoring is one of the many techniques in algebra that is covered by the vague and tricky verb simplify. I can't give you a perfectly clean understanding of this term, since it is inherently a bit vague. It means, in general, to manipulate a mathematical expression to turn it into a form that is easier: easier to understand, easier to work with in the future, easier to compare to other things.

Often, if there is a common factor to a number of terms, it will be easier to understand the expression in factor form. Therefore, often simplify will involve factoring as discussed in this section. It may also involve other techniques -- it's not just factoring.

I wish I could be more concrete about how to simplify, but no one simplification system applies to all situation. How and when to simplify is much more a matter of experience and practice.