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Algebra Chapter 1 Review
The material in this lesson is a review of previous algebra or precalculus
courses. You
should be able to work through this material at a fairly rapid pace. If you find
that much of
it is unfamiliar, or difficult, then you should consider reviewing at greater
length. Any
college algebra text should have material similar to this lesson. At the
University of Kansas,
the current text is College Algebra by Sullivan.
PART 1
By the end of this part of lesson 1, you should be able to
• distinguish between open, closed, and halfopen intervals;
• understand and evaluate expressions involving absolute value; and
• work with exponents, including negative exponents and radical exponents, using
the
Laws of Exponents.
Reading Assignment: Section 1.1 of Chapter 1
Notes and Comments
1. The symbol for absolute value is universal. Pay special attention to the
definition of
the absolute value in the first highlighted box on page 6 of the text. The
Absolute
Value Properties listed in the second highlighted box on page 6 are also very
important. Be sure to go over Example 3 at the bottom of page 6.
2. The Laws of Exponents (text, page 8; Table 1.4) have all the rules that
you will need
when dealing with either integer or fractional exponents. It is important to
remember
that an exponent is only attached to the obvious base; 2t^{2} is not the same as
3. The connection between nth roots of a number and
fractional exponents is a very
important one (page 8; Table 1.3). You should be able to use these two concepts
interchangeably.
4. The distinction between multiplying or dividing two
exponential expressions to the
same base—Laws 1 and 2—and taking powers—Law 3—is important. Multiplying
two exponential powers to the same base turns into addition or subtraction of
exponents, while raising an exponential to a power requires a product or
division of
exponents.
5. Be careful when applying the Laws of Exponents. The
exponent does NOT satisfy the
distributive property. Thus, to multiply (a + b)^{n} one must use the
binomial expansion
to expand this expression .
Practice Assignment
Section 1.1: Exercises 3, 7, 15, 19, 23, 25, 39, 41, 49, 55, 77, 81, 111, 113,
115, 119.
PART 2
By the end of this part of lesson 1, you should be able to
• multiply and factor algebraic expressions, especially
expressions involving
polynomials;
• use the Quadratic Formula to find the roots of polynomials of degree two; and
• simplify algebraic expressions (which may involve all arithmetic operations)
and
arrive at a single fraction involving no negative exponents, and no common
factors
between the numerator and the denominator.
Reading Assignment: Section 1.2 of Chapter 1
Notes and Comments
1. Multiplication of algebraic expressions is
accomplished, in general, by using the
distributive rule illustrated in Example 2 (text, page 13). This method will
always
provide the product, although there are some situations where a commonly used
product has a particular form; see Table 1.5 (page 13).
2. Factoring is the operation that attempts to
unmultiply. This type of operation is
always more difficult than multiplication; see Table 1.6 (page 15). Factoring
cannot
always be accomplished. Probably the best way to start is to attempt to group
the
monomials so that a common factor can be identified. Then, using the
distributive law
in reverse, the original expression can be written as a product.
3. A root, sometimes called a zero, of a polynomial p is a
number a such that p(a) = 0.
The roots of a polynomial are very difficult to find in general, and the problem
of
finding roots is one that goes back to ancient Egypt and Mesopotamia. The
Babylonians and the Greeks could find roots of polynomials for all polynomials
of
degree 1 and 2, but it was in the fourteenth century that roots of polynomials
of degree
3 and 4 could be solved.
4. The Quadratic Formula gives a method for solving all
polynomials of degree 2. This
formula was known by the Greek mathematicians of the B.C. era. While it may be
possible to solve such a polynomial by factoring and then applying the fact that
a
product of real numbers is equal to zero if and only if at least one of the real
numbers
is equal to zero, the Quadratic Formula gives a surefire and quick method for
finding
the roots of a second degree polynomial.
5. Doing addition and subtraction with rational
expressions (quotients of polynomials) is
always difficult. You must remember that these operations can only be
accomplished
by first finding a common denominator, just as when trying to add or subtract
rational
numbers. The explicit rules for this are in Table 1.8 (page 19). In contrast,
multiplication and division are relatively easy to perform. When in doubt,
perform the
desired operations with rational numbers and compare. The rules for multiplying
or
dividing are in Table 1.7 (page 19).
6. Sometimes an algebraic expression will contain several
rational expressions, together
with some of the arithmetic operations. The process for dealing with such
expressions
is the same as the process when rational numbers replace the rational
expressions. In
example 10a (page 20), the problem is to find the quotient of two expressions.
The only
way we can simplify a quotient is to get both the numerator and denominator in
the
form of rational expressions and then use Table 1.7. In example 10b, we first
need to
eliminate the negative exponents, and then proceed in a manner similar to 10a.
7. Cancellation between terms in the numerator and
denominator can be very tricky. The
best way to determine whether cancellation can occur is by factoring both the
numerator and denominator before canceling. Cancellation can only occur if there
are
common factors in both the numerator and denominator. See Example 8 on page 19
of
your text.
Practice Assignment
Section 1.2: Exercises 15, 19, 37, 41, 43, 49, 57, 63, 67, 73, 77, 81, 85, 87.
PART 3
By the end of this part of lesson 1 you should be able to
• compute the distance between two points;
• find the slope of a line; and
• use the pointslope form to find the equation of a straight line.
Reading Assignment: Sections 1.3 and 1.4 of Chapter 1
Notes and Comments
1. Every point in the plane can be associated with a pair
of real numbers and, conversely,
every pair of real numbers can be associated with a point in the plane. It is
this
association—both ways—that allows algebra and geometry to flourish together: a
geometric concept can be given an algebraic interpretation, and an algebraic
expression can be associated with a geometric figure in the Cartesian plane.
(The word
Cartesian is in honor of Rene Descartes, who with Pierre Fermat invented
analytic
geometry.)
2. The Euclidean, or Cartesian, plane arises from having
two intersecting number lines.
Their point of intersection is called the origin, and is denoted by O. Since the
two lines
are number lines, each has a scale of units, although the scales are not
necessarily the
same on both lines. Whenever we work with analytic geometry, for example in
drawing graphs , these scales must be clearly denoted. There are a few
conventions
that have arisen concerning these number lines. For example, the number lines
intersect in a right angle; one horizontal—called the xaxis—and the other
vertical—called the yaxis. On the xaxis, positive reals are displayed to the
right,
negatives to the left, while on the yaxis, positives go up and negatives down.
Because
the xaxis and yaxis intersect at right angles to each other, we can use the
Pythagorean
theorem in order to compute the distance between any two points. The reason for
this
is that any vertical line and any horizontal line must also intersect in right
angles.
Therefore, the line connecting the two points, together with a line parallel to
the xaxis
and another parallel to the yaxis, form a rightangle triangle. (See Figure
1.7, page 25.)
3. Every line in the Euclidean plane, except vertical
lines, has a slope. This slope is
computed by selecting two points on the line and using Equation 3 on page 33.
Note
that it does not matter what two points you choose; the slope will always be the
same.
There are two important slopes that you should remember. The slope of a vertical
line
is undefined. Why? Horizontal lines have slopes equal to zero.
4. One of the two parts of calculus, differential
calculus , deals with tangents to the graph
of a function. In the situations we will encounter, we will be given a point on
the
graph and the slope of the tangent line. Thus, while there are several formulas
for
determining the equation for a (tangent) line, the pointslope form is the most
important and you should use this form when doing the exercises.
5. Observe that in the distance formula, it is unimportant
whether we use (a − b) or
(b − a) in computing the distance, but in computing the slope, confusing the
designation of the first versus the second point may result in an error in the
sign of the
slope . Thus, the slope of the line through the points (2, 4) and (−3, 6) is
given by
6. Lines with positive slopes will rise from left to
right, while lines with negative slopes
will fall from left to right. Parallel lines have equal slopes and two lines
that are
perpendicular (intersect at right angles) will have slopes such that their
product equals
(−1) (page 37).
7. In the slope intercept form y = mx + b, m is the slope
and the point (0, b), is the
yintercept—the point where the line intersects the yaxis. If the equation is
given in
this form, or can be algebraically manipulated to get this form, then the slope
of the
line will be given by the coefficient of x . However, it is crucial that the
equation be
written so that the coefficient of y equals 1.
Practice Assignment
Section 1.3: Exercises 21, 23, 25, 27, 29, 35, 39.
Section 1.4: Exercises 13, 15, 17, 23, 27, 41, 51, 53, 69, 71.
Written Assigment 1
Section 1.1: Exercises 26, 56, 82, 90, 114, 116.
Section 1.2: Exercises 6, 10, 42, 58, 74, 84.
Section 1.3: Exercises 26, 28, 30, 34, 36, 38.
Section 1.4: Exercises 14, 16, 18, 42, 54, 70, 76.
Don’t forget:
• If an exercise asks you to graph a particular function,
this graph should be included as
part of the solution to the exercise.
• Every graph must have a scale of units clearly marked on both axes.
• Be sure to show all necessary work, or explain completely the process used to
find the
answer to an exercise.
• Answers obtained by using a calculator must be correct to three (3) decimal
places.
• Be sure to include an Independent Study cover sheet with your assignment.
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