1. Algebra 1.1 Basic Algebra 1.2 Equations and Inequalities 1.3 - - PowerPoint PPT Presentation

• 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 Fractions

Fractions are essential to mathematics

• Adding fractions:
• Multiplying fractions:
• Improper fractions:

a b + c d = ad + bc bd a b · c d = ac bd

a b c d

= ad bc

Compute: 12 17 + 2 3

Compute: x2 y7 + y3 x4

Powers and Roots

For any positive whole number n, xn = x · x · ... · x

| {z }

n times

If , we say is an root of .

y = xn x nth y

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

nth

n

√x

x

1 n

Notice that:

⇣ x

1 n

⌘n =x

1 n ·n

=x1 =x

Simplify:

4

√ 16 34

Solve for x : x

3 2 = 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 = x2 f(x) = x − 1 f(x) = x6 − x + 1 y = x2 + 2x − 1 f(r) = 4r2 − 9

First Order Polynomials

These are just lines:

y = ax + b

Second Degree Polynomials

y = ax2 + 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) = acx2 + (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 x2 + 4x + 3

Roots of Quadratics

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

ax2 + bx + c = 0 ⇔ x = −b ± √ b2 − 4ac 2a

Find Roots of x2 − 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 ( )

xy 2x x2 2x x2

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 x0 = 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) ⇔ ay = x

Properties of Logarithms

Logarithms 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

loga(ax) = aloga(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

• f 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 = ax2 + 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 = ax2 + bx + c ⇔ x = −b ± √ b2 − 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

• r A ≤ 0 and B ≤ 0

ax2 + 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

• r 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.

x2 + bx + c ≥ 0, x2 + bx + c = (x − α)(x − β)

Solve for x : x2 − 6x + 5 ≤ 0

Rates of Change of Quadratic Functions

• A useful characterization
• f 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) = x3

Plot f(x) = −x4 + 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 (x2 − 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 = ax ⇔ loga(y) = x

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

Solve for x : 4x−1 = 16

Solve for x : log2(x) = 3

Solve for x : ex > e2x

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 Equations

The 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

• n 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 = x2 + x + 1

While more complicated looking, this system can be solved with our substitution method, combined with the quadratic formula.

Systems of Inequalities

One 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

     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      x + z = 1 2x + 2y = 1 3y − z = 2