# Algebra Chapter 1 Review

The material in this lesson is a review of previous algebra or pre-calculus
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 half-open 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
un-multiply. 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 sure-fire 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 point-slope 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 x-axis—and the other

vertical—called the y-axis. On the x-axis, positive reals are displayed to the
right,

negatives to the left, while on the y-axis, positives go up and negatives down.
Because

the x-axis and y-axis 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 x-axis

and another parallel to the y-axis, form a right-angle 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 point-slope 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

y-intercept—the point where the line intersects the y-axis. 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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