The purpose of this section is for readers to review important algebraic concepts. It is necessary to understand algebra in order to do calculus. If you are confident of your ability, you may skim through this section.
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. However, 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 ."
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, where .
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)
exactly if and , , and .
Logarithms are taken with respect to some base. What the equation is saying is that is the exponent of that will give you .
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.
Most scientific calculators have the and functions built in., which do not include logarithms with other bases. Consider how one might compute , where and are given known numbers, when we can only compute logarithms in some base . First, let us assume that
Then the definition of logarithm implies that
If we take the base log of each side, we get
Solving for , we find that
For example, if we only use base 10 to calculate , 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 of and , respectively. Applying the previous rule we have
The quadratic formula
Given any quadratic equation , all solutions of the equation are given by the quadratic formula:
Note that the value of will affect the number of real solutions of the equation.
There are two real solutions for the equation
There are only one real solutions for the equation
There are no real solutions for the equation
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
Vieta's formulae relate the coefficients of a polynomial to sums and products of its roots. It is very convenient because under certain circumstances when the sums and products of the quadratic's roots are provided, one does not require to solve the whole quadratic polynomial.
Vieta's formulae in quadratic polynomials
Given any quadratic equation , The roots of the quadratic polynomial satisfy
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 .
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 and denominator has degree with .
Suppose that divided by has quotient and remainder . That is