1. Algebra

1.1 Basic Algebra

1.2 Equations and Inequalities

1.3 Systems of Equations

1.1 Basic Algebra

1.1.1 Algebraic Operations

1.1.2 Factoring and Expanding Polynomials

1.1.3 Introduction to Exponentials

1.1.4 Logarithms

1.1.1 Algebraic Operations

We need to learn how our basic algebraic operations interact. When confronted with many operations, we follow the order of operations:

Parentheses Exponentials Multiplication Division Addition Subtraction

Simplify the following expression

2 ⇤ (2 + 4)(3�2)

Manipulating FractionsFractions are essential to mathematics

• Adding fractions:

• Multiplying fractions:

• Improper fractions:

a

b+

c

d=

ad+ bc

bd

a

b· cd=

ac

bd

abcd

=ad

bc

Compute:

12

17

+

2

3

Compute:

x

2

y

7+

y

3

x

4

Powers and RootsFor any positive whole number n, x

n = x · x · ... · x| {z }n times

If , we say is an root of .y = x

nx nth y

Roots undo powers, and vice versa. We denote roots as or . nth n

px

x

1n

Notice that:⇣x

1n

⌘n

=x

1n ·n

=x

1

=x

Simplify: 4p16

34

Solve for x : x

32= 81

1.1.2 Factoring and Expanding Polynomials

Polynomials• Polynomials are functions

that are sums of nonnegative integer powers of the variables.

• The highest power is called the degree of the polynomial.

• Higher degree polynomials are generally harder to understand.

y = x

2

f(x) = x� 1

f(x) = x

6 � x+ 1

y = x

2 + 2x� 1

f(r) = 4r2 � 9

First Order Polynomials

These are just lines:

y = ax+ b

Second Degree Polynomials

y = ax

2 + bx+ c

We can try to factor quadratics, i.e. write as a product of first order polynomials.

Expanding Quadratics(a+ b)(c+ d) 6= ac+ bd

(a+ b)(c+ d) = ac+ ad+ bc+ bd

(ax+ b)(cx+ d) = acx

2 + (ad+ bc)x+ bd

One can undo factoring by expanding products of polynomials. One must take care with distribution.

Expand (x+ 3)(�2x� 1)

Factor x

2+ 4x+ 3

Roots of Quadratics

These can be found by factoring, and also with the famous quadratic formula:

ax

2 + bx+ c = 0 , x =�b±

pb

2 � 4ac

2a

Find Roots of x

2 � x� 5

Higher Order Polynomials• These can also be

factored, though it is usually harder.

• Formulas like the quadratic formula exist for degree 3,4, polynomials.

• Nothing for degree 5 and higher.

1.1.3 Introduction to Exponentials

• The number x is called the base.

• The number y is called the exponent.

• Examples include: (base x, exponent 2) and (base 2, exponent x)

• When someone refers to an exponential function, they mean the variable is in the exponent (i.e. ), not the base ( )

x

y

2xx

2

2x

x

2

Properties of Exponents

Basic Rules:

• (same base, different exponents)

• (different base, same exponent)

• (iterated exponents)

• for any value of x (convention)

ax+y = axay

axbx = (ab)x

(ax)y = axy

x

0 = 1

1.1.4 Logarithms

• We call a the base.

• Logarithms are a compact way

to solve certain exponential

equations:

y = loga(x)

y = loga(x) , a

y= x

Properties of LogarithmsLogarithms enjoy certain algebraic properties, related to the exponential properties we have already studied.

• (logarithm of a product)

• (logarithm of a quotient)

• (logarithm of an exponential)

• (logarithm of 1 equals 0)

loga(xy) = loga(x) + loga(y)

loga

✓x

y

◆= loga(x)� loga(y)

loga(xy) = y loga(x)

loga(1) = 0

loga(a) = 0

Logarithm as Inverse of Exponential

log

a

(a

x

) = a

loga(x)= x

1.2 Equations

1.2.1 Linear Equations and Inequalities

1.2.2 Quadratic Equations

1.2.3 Higher Order Polynomials

1.2.4 Exponential and Logarithmic Equations

1.2.5 Absolute Value Equations

1.2.1 Linear Equations and Inequalities

Equations of the form

We want to compute values of given and vice versa.

Sometimes we need to perform some algebraic rearrangements first.

y = ax+ b

x

y

Solve for x : 2 = 4x� 3

Linear Inequalities

Linear equations can be broadened to linear inequalities of the form , with potentially in place of .

Since defines a line in the Cartesian plane, linear inequalities refer to all points on one side of a line, either including ( ) or excluding ( ) the line itself.

y mx+ b

�, <,>

y = mx+ b

,� <,>

Solve for x : 3x� 3 1

1.2.2 Quadratic Equations

Quadratic refers to degree two polynomials. Quadratic equations are equations involving degree two polynomials:

y = ax

2 + bx+ c

Unlike linear equations, in which simple algebraic techniques were sufficient, finding solutions to quadratics requires more sophisticated techniques, such as:

• Factoring • Quadratic Formula • Completing the Square

Quadratic Formula

A formulaic approach to solving quadratic equations is the quadratic formula:

0 = ax

2 + bx+ c , x =�b±

pb

2 � 4ac

2a

In particular, quadratic equations have two distinct roots, unless .b2 � 4ac = 0

Quadratic Inequalities

Solving quadratic inequalities can be made easier with the observation that

AB � 0 ,A � 0 and B � 0

or A 0 and B 0

ax

2 + bx+ c � 0

This suggests factoring our quadratic, and examining when each linear factor is positive or

negative.

Similarly, AB 0 ,A � 0 and B 0

or A 0 and B � 0

Again, we see that if we can factor our quadratic into linear factors, we can examine each factor

individually. Indeed, supposing that our quadratic inequality has the form

we can factor and examine the corresponding linear factors.

x

2 + bx+ c � 0,

x

2 + bx+ c = (x� ↵)(x� �)

Solve for x : x

2 � 6x+ 5 0

Rates of Change of Quadratic Functions

• A useful characterization of quadratic polynomials is that their rate of change is a linear function.

• This is an early result in calculus, but we will not prove it.

1.2.3 Higher Order Polynomials

• One can also consider polynomials of degree higher than 2.

• These are generally harder to analyze and plot.

• One can use basic heuristics, however, in addition to graphing calculators.

• Odd degree polynomials have the two “tails” pointing in opposite directions.

• Even degree polynomials have the two “tails” pointing in the same direction.

• Odd degree polynomials always have at least one (real) root, while even polynomials need not.

Plot f(x) = x

3

Plot f(x) = �x

4+ 4

Number of Roots

• A degree polynomial has at most distinct real roots.

• It has has at most minima and maxima.

n

n

n� 1

Find all roots of (x

2 � 4)(x+ 2)(x� 1)

2

1.2.4 Exponential and Logarithmic Equations

These may look daunting! However, we can use our exponential and logarithmic properties (tricks) to make our lives easier; see Lecture 1.3,1.4.Recall that . y = a

x , log

a

(y) = x

From this, we can approach many equations that look intimidating.

Solve for x : 4

x�1= 16

Solve for x : log2(x) = 3

Solve for x : e

x

> e

2x

1.2.5 Absolute Value Equations

Recall the absolute value function, which is equal to a number’s distance from 0:

|x| =(x, x � 0

�x, x < 0

In other words, the absolute value function keeps positive numbers the same, and switches negative numbers into their positive counterpart.

Equations with Absolute Value

When considering equations of the form:

it suffices to consider the two cases

|f(x)| = g(x)

f(x) = g(x) and � f(x) = g(x)

In the case of absolute value equations involving first order polynomials (linear functions), we get:

|ax+ b| = c , ax+ b = c or � (ax+ b) = c

Solve for x : |2x� 5| = 1

Inequalities Involving Absolute Values

When considering systems of absolute value inequalities, great care must be taken.

In general,

|f(x)| g(x) ,(f(x) g(x) and f(x) > 0

�f(x) g(x) and f(x) 0

A similar equivalence holds for |f(x)| � g(x)

Linear Absolute Value Inequalities

One can, when working with inequalities of the form

proceed by finding the two solutions to

then plotting these on a number line, and checking in which region the desired inequality is achieved. This is the number line method.

|ax+ b| c or |ax+ b| � c

ax+ b = c and � (ax+ b) = c

Solve for x : |x+ 2| = 5

1.3 Systems of Equations

1.3.1 Systems of Equations and Inequalities

1.3.2 Higher Order Systems

1.3.1 Systems of Equations and Inequalities

A classic area of mathematics is solving two or more systems of

equations or inequalities simultaneously.

On classic formulation is: find the intersection of two lines, given their

equations

System of Linear EquationsThe problem of finding the intersection of two lines may formulated as the algebraic problem of finding the simultaneous solution to system of linear equations (

y = m1x+ b1

y = m2x+ b2

Classical solution method: Set the two expressions on the right equal and solve for , then go back and solve for

x

y.

Solve :

(�3x+ 4 = y

4x� 7 = y

It is possible to mix other types of equations into systems. The same techniques as

before work.

(y = 3x+ 4

y = x

2 + x+ 1

While more complicated looking, this system can

be solved with our substitution method, combined with the

quadratic formula.

Systems of InequalitiesOne can also study regions in the Cartesian

plane in which a inequalities are simultaneously satisfied.

In the case of linear inequalities, these may be of the form:

(y m1x+ b1

y m2x+ b2

(y � m1x+ b1

y m2x+ b2

(y � m1x+ b1

y � m2x+ b2

Solve :

(�2x+ 1 y

x+ 2 y

1.3.2 Higher Order Systems

• We have so far considered systems of equations with two variables and two unknowns.

• In general, it is possible to consider more equations and more unknowns.

• On the CLEP exam, it is good to know how to solve linear systems with three variables and three unknowns.

• These look like: for coefficients 8><

>:

a1x+ b1y + c1z = d1

a2x+ b2y + c2z = d2

a3x+ b3y + c3z = d3

ai, bi, ci, di, i = 1, 2, 3

• These systems can be solved by Gaussian elimination a.k.a. row reductions.

• Basically, equations are added and subtracted to isolate variables. It is long and tedious.

• For CLEP questions, checking each purported solution may be more time-efficient than directly solving the system.

Solve

8><

>:

x+ z = 1

2x+ 2y = 1

3y � z = 2