October 30th, 2006 . by Vanaja
**Remainder Theorem**

Let *p(x)* be any polynomial of degree n>0 ,and *a* any real number. If* p(x)* is divided by *( x-a),* then the remainder is *p(a).*

The Remainder Theorem can be proved as follows.

**Proof**

Let us suppose when *p(x)* is divided by *( x-a),* the quotient is *q(x)* and remainder is* r(x).*

So we have,

*p(x)=(x-a) q(x)+r(x),* where *r(x)=*0 or degree of *r(x)< *degree of* x-a.*

Since degree of *( x-a)* is1, either *r(x)=*0 or degree of *r(x)=*0

So *r(x)* must be a constant,say r.

Thus for all values of x,

*p(x)=(x-a) q(x)+r* ……..(1), where r is a constant.

In particular, when x=a,

*p(a)=0.q(x)+r*

=0+r

=r

Hence the theorem.

**Factor Theorem**

Let *p(x)* be a polynomial of degree n>0. If* p(a)=0* for a real number a, then* (x-a)* is a factor of

*p(x).* Conversely, if *(x-a)* is factor of *p(x),* then *p(a)=*0.

**Proof **

*First part:*

Let p(a)=0

Then by remander theorem, r=0

So equation (1) becomes

*p(x)=(x-a) q(x)*

==>(x-a) is a factor of p(x).

*Second Part:*

By remainder theorem,

*p(x)=(x-a) q(x)+r*

ie. *p(x)=(x-a) q(x)+p(a)*

Since (x-a) is a factor, p(a) must be zero.

This proves the theorem.

**Example**

Find the remainder when p(x)= x^2 +3x+1 is divided by x+1.

Determine whether( x-2) is a factor of p(x) or not.

**Solution**

x+1= x-(-1)

[*Always check whether the divisor is in the form of (x-a)or not. Otherwise rewrite that in the form of(x-a)*]

So, here a=-1

There fore by remainder theorem, the required remainder is

p(a)= p(-1)

=(-1)^2+3(-1)+1

=1-3+1

=-1

We know, by factor theorem,

if (x-2) is a factor of p(x), then p(2) must be zero.

Here p(2)=2^2+3(2)+1

=4+6+1

=11 which is not equal to zero.

So (x-2) is not a factor of p(x).

Posted in Algebra ** | **
1 Comment »

October 29th, 2006 . by Vanaja
Today I received an interesting E-Mail. Here is the Mail and the reply I have sent.

Plot the point with coordinates (3, –5).

Graph 2x – y = 4.

Graph using the intercept method: 2x + y = 4.

Write the equation of the line with slope 4 and y-intercept (0, –5). Then

graph the line.

Math For You to P

Hi P

Thanks for writing to me

I think you have four questions. So my fee willbe $6. If that is O.K for you, I will send the detailed solutions for you.

P to me

You don’t have to send me a solution for my problems because the site said

that it was free.

Math For You to P

No. Where did you see that? I have clearly mentioned about my service at the home page.Here also you can see more details. #Why did you send the questions then?

P to me

Like I mentioned earlier the site said something about getting free help

witrh your math. I got the site from another math student so maybe I

clicked on the wrong site. That is why I posted the questions. Once again

you can disregard the questions and I will found help elsewhere. Sorry if

I wasted your time.

Math For You to P

No P,

You came from MSN for the keyword ‘Math Genius’.(http://search.msn.com/results.aspx?q=math ) Not even for math help.So you are not a targeted visitor for me.So it do not worry me.(otherwise also no problem.I have been getting enough clients.) You are from Bastrop,Louisiana, U.S.You go to anywhere for help.(are you really a student?)That is not my topic. But I have a humble request : Don’t lie.I have the detailed track record of your visit.

Don’t worry.Take a deep breath and Relax.

P to me

I think I misread the article it said free lessons. Like I say disregard

the questions because I already have the solution.

Math For You to P

That is O.K. No problem.

Posted in Uncategorized ** | **
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October 27th, 2006 . by Vanaja
Two trains start from two different stations A and B. One is going from station A to station B and another from station B to station A. If first train takes 8 hours to complete the journey and the second one takes 12 hours, when do the two trains cross each other?

**Answer**

The first and second train travel 1/8 and 1/12 respectively of the total distance of AB in one hour . So they cover 1/8 +1/12=5/24 of the distance AB in one hour. That means they will cross each other after 24/5 (4 hours 48 minutes) hours.

Posted in Puzzles ** | **
4 Comments »

October 24th, 2006 . by Vanaja
Given a point *l *and a point P. If P is an element of *l*,

then we say that P lies on *l*, or

P is incident on *l*, or

*l* passes through P.

Now we can try to find answers to the following questions on the basis of our experience.

- Given a point P, is there a line that passes through P? how many such lines are there?
- Given two distinct points A and B, is there a line that passes through both a and B? How many such lines are there?
- Given a line
*l*, is there a point that lies on it? How many such points are there?
- Given two distinct lines
*l *and* m*, is there a point that lies on both* l* and *m*? how many such points are there?

Based on the results of the above questions we can arrive at the following conclusions. These conclusions have to be taken as axioms.

**Incidence Axiom 1:** A line contains infinitely many points.

**Incidence Axiom 2:** Through a given point, there pass infinitely many lines.

**Incidence Axiom 3:** Given two distinct points A and B, there is one and only one line that contains both the points.

According to the third axiom, any two distinct points of the plane determine a line uniquely and completely.

**Definition:** Three or more than three points are said to be collinear, if there is a line which contains them all.

**Definition:** Three or more than three lines are said to be concurrent if there is a point which lies them all.

Posted in Geometry ** | **
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October 20th, 2006 . by Vanaja
Yesterday I couldn’t write. Today I hoped to write about basic geometric concepts. But today also I am very busy and I haven’t enough time to write a long topic.

So we can have a small puzzle today. This is an ancient problem dating back to about 310 A.D.

Demochares had one-fourth of his life as a boy, one-fifth as a youth , one-third as a man, and has spent 13 years in his dotage. How old was Demochares?

**Another problem of this kind:**

A stone weights one Kilogram and half of its weight. What is the actual weight of the stone?

Solutions:

First Question

Let x be Demochares’s age.

Then, x/4+x/5+x/3+13=x

ie 47x/ 60+13=60

==>47x/60=47

==>x=60

So, Demochares was **60** years.

Second Question

**2 kilogram**

Posted in Puzzles ** | **
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October 18th, 2006 . by Vanaja
It is very important for a math student to learn the basic concepts in mathematics. It is a common fact that most children find math is very hard and in particular Geometry. The main reason behind this is that they don’t have the basics in maths.

and in Geometry the concepts are more abstract. If the students get some good basic help in math, I am sure most of them do better in maths. So let us learn some basics of Geometry today.

There are three basic concepts of geometry. These are “**point**“, “**line**” and “**plane**“. I am not attempting to define them as it is not possible to define them precisely. We can however, have a good idea of these three by considering examples. A fine dot made by a sharp pencil on a sheet of paper, resembles a geometrical point very closely. The sharper the pencil, the closer is the dot to the concept of a point.

The surface of a sheet of paper or the surface of a smooth table are examples of plane. But these surfaces limited in extent. The geometrical plane extends endlessly in all directions.

A straight line, drawn on a sheet of paper with a sharp pencil, is a close example of a geometrical straight line. A geometrical line is a set of points and extends endlessly in both the directions. To emphasize this we use two arrowheads.

It is impossible to find exact example for point,line and plane in nature. The geometrical point, line and plane are ideal concepts. but for practical purpose it is enough to deal with close examples.

So, a plane is a set of points, line is a subset of plane. Moreover all the other figures in geometry are sets of points. But they are not just set of points. They are special set of points possessing some properties.

**Notation**

We use capital letters such as A, B, C, P, Q, R, X, Y, Z etc. to denote points.

We use small letters (lower case) such as l,m,n,p,q,r etc. to denote lines.

Posted in Definitions, Geometry, Math Articles ** | **
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October 17th, 2006 . by Vanaja
**What is the peculiarity of 1729?**

This number is known in the name of the famous mathematician Ramanujan.

**Answer**

This is the one and only one number which can be written as the sum of the cubes of two numbers in two different ways.

10^3 + 9^3 =1729

12^3 + 1^3 = 1729

Posted in General, Puzzles ** | **
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October 16th, 2006 . by Vanaja
I can help you in the following topics of Mathematics

You can send in me your math problems at **[email protected]**

**All math solutions will be clearly demonstrated.**

**Algebra **

Middle school and high school algebra

Functions

Linear Equations, Inequalities.

Quadratic Equations

Simultaneous System of Equations

Arithmetic and Geometric Progressions.

Sequences and Series

Binomial Theorem

Probability, Permutations, Combinations.

Logarithms

Matrices and Determinants

**Trigonometry**

Trigonometric Identities.

Trigonometric Functions

Solutions of Triangles

Heights and Distance Problems

**Geometry **

Middle school and High school Geometry

Prisms

Similar triangles,

Congruent Triangles

**Analytic Geometry**

The Straight Line.

The Circle.

Conic Sections (Parabola,Ellipse,Hyperbola)

**Statistics **

Probability

Mean Deviation

Standard Deviation

**Calculus**

Functions, Limits and Continuity.

Differentiation

Integration

Definite Integral

Definite Integral Applications (Areas and Volumes)

Posted in Math Homework Help ** | **
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October 16th, 2006 . by Vanaja
Today again I have a puzzle. This one is not a typical math puzzle. You should use your common sense to answer this simple puzzle. Here we go……

Two identical trains, at the equator start travelling round the world in opposite directions. They start together, run at the same speed and are on different tracks.

Which train will wear out its wheel treads first?

**You can post your answers in the comments section.**

Update

The train travelling against the spin of the earth will wear its wheel more quickly, as the centrifugal force is less in this train.

Posted in Puzzles ** | **
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October 14th, 2006 . by Vanaja
I offer math tutoring via email.This is a great option for students looking for an online tutor, or students who don’t necessarily need to sit down with a tutor for a long time. This can also be a good option for homeschooling students and those working in independent study situations.

For a long session, I will likely ask you to email me a sample or two of the types of questions you have, so that I can be sure that it looks like something I can help you with effectively and I can calculate my fee.Once you agreed to puchase my service,you can pay me through my Paypal account **[email protected]**. You do not need a Paypal account in order to use PayPal, as long as you have a credit card. (If you are under 13 years, you can ask your parents to open an account.)

**IMPORTANT NOTE**

It is highly recommended that, when you e-mail your questions to me, you include the area of study , the topic of study and your grade level.

Please feel free to describe all the difficulties you’ve been having and what you hope to achieve with tutoring.

Posted in Math Homework Help ** | **
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