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.
Rules of arithmetic and algebra
The following laws are true for all
whether these are numbers, variables, functions, or more complex expressions involving numbers, variable and/or functions.
Addition
- Commutative Law:
.
- Associative Law:
.
- Additive Identity:
.
- Additive Inverse:
.
Subtraction
- Definition:
.
Multiplication
- Commutative law:
.
- Associative law:
.
- Multiplicative identity:
.
- Multiplicative inverse:
, whenever ![{\displaystyle a\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f)
- Distributive law:
.
Division
- Definition:
, whenever
.
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.
Interval notation
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:
Endpoint conditions
|
Inequality notation
|
Interval notation
|
Including both 2 and 4
|
all satisfying
|
|
Not including 2 nor 4
|
all satisfying
|
|
Including 2 not 4
|
all satisfying
|
|
Including 4 not 2
|
all satisfying
|
|
In general, we have the following table:
Meaning
|
Interval Notation
|
Set Notation
|
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
|
|
|
All values
|
|
|
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
.
Exponents and radicals
There are a few rules and properties involving exponents and radicals that you'd do well to remember. As a definition we have that if
is a positive integer then
denotes
factors of
. That is,
If
then we say that
.
If
is a negative integer then we say that
.
If we have an exponent that is a fraction then we say that
. In the expression
,
is called the index of the radical, the symbol
is called the radical sign, and
is called the radicand.
In addition to the previous definitions, the following rules apply:
Rule
|
Example
|
|
|
|
|
|
|
|
|
|
|
Simplifying expressions involving radicals
We will use the following conventions for simplifying expressions involving radicals:
- Given the expression
, write this as ![{\displaystyle {\sqrt[{c}]{a^{b}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5895c24b750d771f4ef3e0316fc03b4154c2e0f)
- No fractions under the radical sign
- No radicals in the denominator
- The radicand has no exponentiated factors with exponent greater than or equal to the index of the radical
Example: Simplify the expression ![{\displaystyle \left({\frac {1}{8}}\right)^{\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4e8879eaee9a0288a78918c134559ee6148233)
Using convention 1, we rewrite the given expression as
(1)
The expression now violates convention 2. To get rid of the fraction in the radical, apply the rule
and simplify the result:
(2)
The resulting expression violates convention 3. To get rid of the radical in the denominator, multiply by :
(3)
Notice that . Since the index of the radical is 2, our expression violates convention 4. We can reduce the exponent of the expression under the radical as follows:
(4)
|
Exercise
Solution
Logarithms
Consider the equation
(5)
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:
The result of applying the operation to both sides of the equation is
(6)
Logarithms are taken with respect to some base. What equation 6 is saying is that
is the exponent of
that will give you
.
Example
Example: Calculate ![{\displaystyle \log _{10}100000}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0925c0f3b720a5b3e298e1327bdbbe2c95bfbbdd)
|
Common bases for logarithms
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.
Properties of logarithms
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.
Converting between bases
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
.
Factoring and roots
Given the expression
, one may ask "what are the values of
that make this expression 0?" If we factor we obtain
![{\displaystyle x^{2}+3x+2=(x+2)(x+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fae9f5c8998196290b286933d875fe65904dbb4)
.
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
![{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d52e4cff7a7157a34586d9a29df412a7d37f574)
.
For example, consider
. On initial inspection we would see that both
and
are squares (
and
) . Applying the previous rule we have
![{\displaystyle 4x^{2}-9=(2x+3)(2x-3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca15e86f5e4ed4404bc9ead60287b975c31b4179)
.
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 ![{\displaystyle 4x^{2}+7x-2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b62d68d6091356eae593e4ad5ca24f9cdcde792)
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 ![{\displaystyle 4x^{2}+7x-2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b62d68d6091356eae593e4ad5ca24f9cdcde792)
|
Note that if
then the roots will not be real numbers.
Simplifying rational expressions
Consider the two polynomials
![{\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2f1f5c0a0a21eacc320128aa5db7c56a0161fc)
and
When we take the quotient of the two we obtain
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,
, from something we didn't.
Here are some formulas that can be quite useful for solving polynomial problems:
Polynomial Long Division
Suppose we would like to divide one polynomial by another. The procedure is similar to long division of numbers and is illustrated in the following example:
Example
Divide ![{\displaystyle x^{2}-2x-15}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f16511ef786d2a244c9d8d6fe4854b673cf4ecb7) (the dividend or numerator) by ![{\displaystyle x+3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b799ed61638e20ff904ab2b65a8564f4e27a1f) (the divisor or denominator)
Similar to long division of numbers, we set up our problem as follows:
![{\displaystyle {\begin{array}{rl}\\x+3\!\!\!\!&{\big )}\!\!\!{\begin{array}{lll}\hline \,x^{2}-2x-15\end{array}}\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7754c3e776f67c5eabc2799baea0f234db27b65)
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:
![{\displaystyle {\begin{array}{rl}&~~\,x\\x+3\!\!\!\!&{\big )}\!\!\!{\begin{array}{lll}\hline \,x^{2}-2x-15\end{array}}\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8822c88a3ba1a571acd1fa7da4c51885a246c300)
, and we multiply by and write this below the dividend as follows:
![{\displaystyle {\begin{array}{rl}&~~\,x\\x+3\!\!\!\!&{\big )}\!\!\!{\begin{array}{lll}\hline \,x^{2}-2x-15\end{array}}\\&\!\!\!\!-{\underline {(x^{2}+3x)~~~}}\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6435b56ed99c7a74a5cde084638cc20beb0942b1)
Now we perform the subtraction, bringing down any terms in the dividend that aren't matched in our subtrahend:
![{\displaystyle {\begin{array}{rl}&~~\,x\\x+3\!\!\!\!&{\big )}\!\!\!{\begin{array}{lll}\hline \,x^{2}-2x-15\end{array}}\\&\!\!\!\!-{\underline {(x^{2}+3x)~~~}}\\&\!\!\!\!~~~~~~-5x-15~~~\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03dd099d1c251a7c01bdf23619f03c10fd906677)
Now we repeat, treating the bottom line as our new dividend:
![{\displaystyle {\begin{array}{rl}&~~\,x-5\\x+3\!\!\!\!&{\big )}\!\!\!{\begin{array}{lll}\hline \,x^{2}-2x-15\end{array}}\\&\!\!\!\!-{\underline {(x^{2}+3x)~~~}}\\&\!\!\!\!~~~~~~-5x-15~~~\\&\!\!\!\!~~~-{\underline {(-5x-15)~~~}}\\&\!\!\!\!~~~~~~~~~~~~~~~~~~~0~~~\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94e3928c5515723874f5a394f5020534b7397aad)
In this case we have no remainder.
|
Application: Factoring Polynomials
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
.
Exercise
2. Factor
![{\displaystyle x-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1a88d34243b98b57c4df9db5724f61b59a4b9d)
out of
![{\displaystyle 6x^{3}-4x^{2}+3x-5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/787369538bb08f137c38101cbc8bc1066b0d99d0)
.
![{\displaystyle (x-1)(6x^{2}+2x+5)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8e5a2351d7d9f00ab7659ed92a8e92e6546fc2)
![{\displaystyle (x-1)(6x^{2}+2x+5)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8e5a2351d7d9f00ab7659ed92a8e92e6546fc2)
Solution
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
and denominator has degree
with
.
Suppose that
divided by
has quotient
and remainder
. That is
![{\displaystyle N(x)=D(x)Q(x)+R(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cda1bdd9a1b5c944d46f2520626c23d2ab9b9243)
Dividing both sides by
gives
![{\displaystyle {\frac {N(x)}{D(x)}}=Q(x)+{\frac {R(x)}{D(x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eee7c31af01ebb745fe7ccade1a2a183429ed8d)
will have degree less than
.
Example
Write ![{\displaystyle {\frac {x-1}{x-3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4db5156b1a4e0ae2648a33cc73debe07136536b) as a polynomial plus a rational function with numerator having degree less than the denominator.
![{\displaystyle {\begin{array}{rl}&~~\,1\\x-3\!\!\!\!&{\big )}\!\!\!{\begin{array}{lll}\hline \,x-1\end{array}}\\&\!\!\!\!-{\underline {(x-3)~~~}}\\&\!\!\!\!~~~~~~~~~2~~~\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13ab426ce828962c0cbdd19d97c770b969127615)
so
![{\displaystyle {\frac {x-1}{x-3}}=1+{\frac {2}{x-3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe5e439e048a1e801cedbac50b635fdb4c65ec1)
|