This section is intended to review algebraic manipulation. It is important to understand algebra in order to do calculus. If you have a good knowledge of algebra, you should probably just skim this section to be sure you are familiar with the ideas.
Let's look at an example to see how these rules are used in practice.
(from the definition of division)
(from the associative law of multiplication)
(from multiplicative inverse)
(from multiplicative identity)
Of course, the above is much longer than simply cancelling out in both the numerator and denominator. But, when you are cancelling, you are really just doing the above steps, so it is important to know what the rules are so as to know when you are allowed to cancel. Occasionally people do the following, for instance, which is incorrect:
The correct simplification is
where the number cancels out in both the numerator and the denominator.
There are a few different ways that one can express with symbols a specific interval (all the numbers between two numbers). One way is with inequalities. If we wanted to denote the set of all numbers between, say, 2 and 4, we could write "all satisfying ." This excludes the endpoints 2 and 4 because we use instead of . If we wanted to include the endpoints, we would write "all satisfying ." This includes the endpoints.
Another way to write these intervals would be with interval notation. If we wished to convey "all satisfying " we would write . This does not include the endpoints 2 and 4. If we wanted to include the endpoints we would write . If we wanted to include 2 and not 4 we would write ; if we wanted to exclude 2 and include 4, we would write .
Thus, we have the following table:
Including both 2 and 4
Not including 2 nor 4
Including 2 not 4
Including 4 not 2
In general, we have the following table:
All values greater than or equal to and less than or equal to
All values greater than and less than
All values greater than or equal to and less than
All values greater than and less than or equal to
All values greater than or equal to
All values greater than
All values less than or equal to
All values less than
Note that and must always have an exclusive parenthesis rather than an inclusive bracket. This is because is not a number, and therefore cannot be in our set. is really just a symbol that makes things easier to write, like the intervals above.
The interval is called an open interval, and the interval is called a closed interval.
Intervals are sets and we can use set notation to show relations between values and intervals. If we want to say that a certain value is contained in an interval, we can use the symbol to denote this. For example, . Likewise, the symbol denotes that a certain element is not in an interval. For example .
is called the base and is called the exponent. Suppose we would like to solve for . We would like to apply an operation to both sides of the equation that will get rid of the base on the right-hand side of the equation. The operation we want is called the logarithm, or log for short, and it is defined as follows:
Definition: (Formal definition of a logarithm) is the number such that
The result of applying the operation to both sides of the equation is
When the base is not specified, is taken to mean the base 10 logarithm. Later on in our study of calculus we will commonly work with logarithms with base . In fact, the base logarithm comes up so often that it has its own name and symbol. It is called the natural logarithm, and its symbol is . In computer science the base 2 logarithm often comes up.
Logarithms have the property that . To see why this is true, suppose that , , and . These assumptions imply that , , and . Then by substitution, . By the property of exponents, we know that , so . The equation will only hold if the exponents on each side are equal, so . Applying our assumptions, we have what we wanted to show, namely, .
Historically, the development of logarithms was motivated by the usefulness of this fact for simplifying hand calculations by replacing tedious multiplication by table look-ups and addition.
Another useful property of logarithms is that . To see why, consider the expression . Let us represent the value of this expression with . Then we have . If we make each side of the equation the exponent of , then after simplification we have . Now raise each side of the equation to the power and simplify to get . Now if you take the base log of both sides, you get . Solving for shows that , which is what we wanted to show.
Most scientific calculators have the and functions built in. But what if you want to compute a logarithm for a different base? For example, suppose you want to compute . To answer this specific question, let's first come up with a general formula and then apply it to our specific problem. Consider how one might compute , where and are given known numbers, when we can only compute logarithms in some base . Let the value of be . Then the definition of logarithm implies that . If we take the base log of each side, we get . Solving for , we find that . So for our example, if we use base 10, we get .
Given the expression , one may ask "what are the values of that make this expression 0?" If we factor we obtain
If , then one of the factors on the right becomes zero. Therefore, the whole must be zero. So, by factoring we have discovered the values of that render the expression zero. These values are termed "roots." In general, given a quadratic polynomial that factors as
then we have that and are roots of the original polynomial.
A special case to be on the look out for is the difference of two squares, . In this case, we are always able to factor as
For example, consider . On initial inspection we would see that both and are squares ( and ) . Applying the previous rule we have
The following is a general result of great utility.
The quadratic formula
Given any quadratic equation , all solutions of the equation are given by the quadratic formula:
Example: Find all the roots of
Finding the roots is equivalent to solving the equation . Applying the quadratic formula with , we have:
The quadratic formula can also help with factoring, as the next example demonstrates.
Example: Factor the polynomial
We already know from the previous example that the polynomial has roots and . Our factorization will take the form
All we have to do is set this expression equal to our polynomial and solve for the unknown constant C:
You can see that solves the equation. So the factorization is
Note that if then the roots will not be real numbers.
The ratio of two polynomials is called a rational expression. Many times we would like to simplify such a beast. For example, say we are given . We may simplify this in the following way:
This is nice because we have obtained something we understand quite well, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): x-1
, from something we didn't.
Divide (the dividend or numerator) by (the divisor or denominator)
Similar to long division of numbers, we set up our problem as follows:
First we have to answer the question, how many times does go into ? To find out, divide the leading term of the dividend by leading term of the divisor. So it goes in times. We record this above the leading term of the dividend:
, and we multiply by and write this below the dividend as follows:
Now we perform the subtraction, bringing down any terms in the dividend that aren't matched in our subtrahend:
Now we repeat, treating the bottom line as our new dividend:
We can use polynomial long division to factor a polynomial if we know one of the factors in advance. For example, suppose we have a polynomial and we know that is a root of . If we perform polynomial long division using P(x) as the dividend and as the divisor, we will obtain a polynomial such that , where the degree of is one less than the degree of .
Application: Breaking up a rational function
Similar to the way one can convert an improper fraction into an integer plus a proper fraction, one can convert a rational function whose numerator has degree and whose denominator has degree with into a polynomial plus a rational function whose numerator has degree Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): \nu
and denominator has degree with .
Suppose that divided by has quotient and remainder . That is