Contents

1  General Information
    1.1  News/Calendar
    1.2  Reference text
    1.3  Grading system
    1.4  Math 11 Course Outline
2  Course notes

General Information

0.1  News/Calendar

Math 11 MHW3 Exam Schedule (8:00-10:00 A.M. Rm. 126)

0.2  Reference text

The standard textbook for this course is

Elbridge P. Vance, Modern Algebra and Trigonometry, Third Edition

0.3  Grading system

To determine your final grade, compute your raw score and refer to the grade table:

${\displaystyle \mathrm{Score}=\frac{1}{2}(\mathrm{Avg}.\mathrm{Dept}.\mathrm{Exams})+\frac{1}{3}(\mathrm{quiz}+\mathrm{homework}+\mathrm{prob}.\mathrm{set})+\frac{2}{3}\%(\mathrm{final}\hspace{1em}\mathrm{exam})}$

Final grade	Raw score range
1.0	93-100
1.25	90-92
1.5	87-89
1.75	84-86
2.0	80-83
2.25	75-79
2.5	70-74
2.75	65-69
3.0	60-64
4.0	55-59

0.4  Math 11 Course Outline

Meetings	Topics
1	Distribution of Class cards, Introduction
2	1.1 Sets and Basic Notations, 1.2 Subsets and Couting
	1.3 Operations on Sets
3	2.1 Algebra of counting numbers, 2.2 Additive Inverses
	and Subtraction,
	2.3 Integers and Factorizations
4	2.4 Multiplicative Inverses and Division, 2.5 Rational
	and Irrational Numbers
5	3.1 Addition of Algebraic expressions,
	3.2 Multiplication of algebraic expressions,
	3.3 Division of Algebraic Expressions
6	3.4 Special products, 3.5 Factors and factoring
7	3.6 Simplification of fractions
8	3.7 Addition of fractions, 3.8 Multiplication and
	division of fractions,
	3.9 Integral and Zero exponents
9	Exercises
10	Review
1st Dept. Exam	Dec 17
11	3.10 Rational exponents, 3.11 Radicals
12	3.12 Addition and Subtraction of Radicals,
	3.13 Multiplication and Division of Radicals
13	4.1 Order Axioms for the real numbers,
	4.2 One-dimensional coordinate system
14	4.3 Two-dimensional Coordinate system,
	4.4 Distance and slope formulas
15	5.1 Functions and relations, 5.2 Graphical
	representation of unctions and relations
16	8.1 Linear function, 8.3 Quadratic function
17	8.4 Solution of the quadratic equations, 8.5 Inequalities
18	8.6 Zeroes and Coeeficients of the quadratic equation
19	Exercises
20	Review
2nd Dept. Exam	Feb 11
21	8.7 Equations in Quadratic form,
	8.8 Equations involving radicals
22	8.10 Solution of two linear equations,
	8.11 Solutions of three linear equations
23	8.12 Solution of linear and quadratic
24	10.1 Certain theorems on polynomials, 10.4 rational roots
25	11.1 Inverse functions
26	11.2 Exponential functions, Variation
27	14.2 Geometric progression, 8.2 Arithmetic progression
28	Exercises
29	Review
3rd Dept. Exam	Mar 17
Review
Final Exam

Chapter 1
Course notes

4.1 Order Axioms for the real numbers

Prob. List the basic order axioms for real numbers with their definitions.

Ans.

• Trichotomy axiom
For any $a,b\in R$, one and only one of the following are true:


${\displaystyle a>b,a=b,a<b}$

.
• Transitivity axiom
If $a,b,c$ are in $R$ and if $a>b$ and $b>c$ then $a>c$.
• Addition property
If $a,b,c,$ are in $R$ and if $a>b$, then $a+c>b+c$.
• Multiplication property
If $a,b$ and $c$ are in $R$ and if $a>b$ and $c>0$ then $a·c>b·c$.

Prob. What is the definition of '<' or 'less than' ?

Ans. If $a>b$, then $b<a$.

4.2 One-dimensional coordinate system

4.3 Two-dimensional Coordinate system

4.4 Distance and slope formulas

Prob. Write the distance formula between two points between two points $P(x_1,y_1)$ and $Q(x_2,y_2)$.

Ans.

${\displaystyle d=\sqrt{(x_2-x_1){}^2+(y_2-y_1){}^2}}$

Problem. What is the midpoint $(x_m,y_m)$ of a line segment defined by $P(x_1,y_1)$ and $Q(x_2,y_2)$?

Ans.

${\displaystyle x_m=(x_1+x_2)/2;y_m=(y_1+y_2)/2}$

Prob. Give the formula for the slope $m$ of a line defined by two points $P(x_1,y_1)$ and $Q(x_2,y_2)$.

Ans.

${\displaystyle m=\frac{y_2-y_1}{x_2-x_1}.}$

5.1 Functions and relations

5.2 Graphical representation of functions and relations

8.1 Linear function

Prob. The graph of a linear function is a straight line. List the different forms for expressing the equation of the straight line.

Slope-intercept form	$y=mx+b$
Slope-point form	$(y-y_1)=m(x-x_1)+b$
Two point form	$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$
General implicit form:	$Ax+By+C=0$

8.3 Quadratic function

8.4 Solution of the quadratic equations

Prob. Look at the expansion of perfect squares: $(ax+b){}^2=a^2{x}^2+2abx+b^2$ and consider relabeling the leftmost, middle and rightmost terms so that $L=a^2{x}^2$, $M=2abx$ and $R=b^2$. Express $R$ in terms of $L$ and $M$.

Ans.

${\displaystyle R={\left(\frac{M}{2\sqrt{L}}\right)}^2.}$

Prob. With reference to the previous problem, if the middle term $M$ is missing, determine its value when $L$ and $R$ are given.

Ans.

${\displaystyle M=\frac{\sqrt{\mathrm{LM}}}{2}.}$

Prob. Consider he quadratic equation $f(x)=A{x}^2+Bx+C=0$. Express $f$ in the form $f(x)=A(x+x_0){}^2+K$.

Ans.

Rewrite $f(x)=A(x^2+\frac{B}{A})+C=0$ Let $L=x^2,M=\frac{B}{A}x$. Then $R=\frac{B^2}{4A^2}$. We perform 'completing the square':

${\displaystyle \begin{array}{cc}f(x)& =A(x^2+\frac{B}{A}+\frac{B}{4A^2}-\frac{B}{4A^2})+C\\ & =A(x^2+\frac{B}{A}+\frac{B^2}{4A^2})-\frac{B^2}{4A}+C\\ & =A{(x+\frac{B}{2A})}^2+\frac{4AC-B^2}{4A}\end{array}}$

Therefore, $x_0=\frac{B}{2A}$ and $K=\frac{4AC-B^2}{4A}$

Prob. At what value of $x$ does $f$ has a minimum value when $A>0$? Or determine the global minimum of $f$ when $A>0$.

Ans.

Note that the expression ${(x+\frac{B}{2A})}^2$ is never negative. Thus when $A>0$, $f$ has a global minimum $\frac{4AC-B^2}{4A}$ attained at $x=x_o=\frac{B}{2A}$.

Prob. Determine the global maximum of $f$ when $A<0$.

Ans. Exercise.

Remark. The point $(\frac{B}{2A},\frac{4AC-B^2}{4A})$ is called the vertex of the parabola defined by $f$.

Prob. Derive the quadratic formula

${\displaystyle x_{1,2}=\frac{-B\pm \sqrt{B^2-4AC}}{2A}}$

for obtaining the roots of $f$ where $f(x)=A{x}^2+Bx+C$. This formula is useful when you cannot factor quickly a quadratic expression.

Ans. Exercise.

Prob. What is the nature of the roots of $f=A{x}^2+Bx+C=0$ given the value of the associated discriminant $Q=B^2-4AC$? HINT: Use the quadratic formula.

Ans.

Q	Nature of roots
$>0$	two different real roots
$=0$	two equal real roots
$<0$	no real roots
	two complex conjugate roots

8.5 Inequalities

8.6 Zeroes and Coeficients of the quadratic equation

Prob. Show that for $f(x)={\mathrm{ax}}^2+bx+c$, the sum of the roots $r_1+r_2=-(b/a)$ and the product of the roots is $c/a$.

Ans. Exercise (Hint: Use the quadratic formula).

Prob. What is the general relation between roots and coefficients of a polynomial in $x$?

Ans.

The polynomial $f(x)$ of degree $n$ with leading coefficient $a_0=1$

${\displaystyle f(x)=a_0{x}^n+a_1{x}^{n-1}+\dots +a_n}$

can be expressed in terms of its $n$ roots $r_1,r_2,\dots ,r_n$ as

${\displaystyle f(x)=(x-r_1)(x-r_2)\dots (x-r_n).}$

Now expand the last equation. We obtain the following formulas (attributed to Vieta):

${\displaystyle \begin{array}{c}\\ a_1& =-(r_1+r_2+\dots +r_n)\\ a_2& =+(r_1{r}_2+r_1{r}_3+\dots r_1{r}_n+r_2{r}_3+\dots r_2{r}_n+\dots r_{n-1}{r}_n)\\ a_3& =-(r_1{r}_2{r}_3+\dots +r_1{r}_{n-1}{r}_n+\dots +r_{n-2}{r}_{n-1}{r}_n)\\ \dots & =\dots \\ a_n& =(-1){}^n(r_1{r}_2\dots r_n)\end{array}}$