Zeroes of a quadratic polynomial

Exploring graphs of Quadratic polynomials

A quadratic polynomial is given by ax^2+bx+c , a not equal to zero , a,b,c, arereal numbers. The degree of a quadratic polynomial is 2. There are atmost two zeroes of a quadratic polynomial. Observe the following graphs. This is a graph of a quadratic polynomial which do not interesect the x- axis at any of the points, so there is no zero of this polynomial. In the second graph, the curve touches the x-axis at one point, so it has 1 number of zero. In the following graph, the curve cuts the x-axis at two points, so the given quadratic polynomial has two zeroes. Note: The zero of a quadratic polynomial is the value of x- coordinate of a point where the graph of the polynomial cuts the x-axis. It has atmost 2 zeroes.

Geometrical Representation of polynomials

• Linear Polynomial
  The graph of a linear polynomial of the form ax+b, a not equal to zero is a straight line which intersects the x-axis at exactly one point . The value of x coordinate of the point where the line meets x- axis is zero of the polynomial.

Consider the following graph

This is a grpah of linear polynomial 2x+3. It intersects the x-axis at point (-1.5,0) .

So, the linear polynomial 2x+3 has one zero given by -1.5 (x-coordinate of point where line cuts the x-axis)

Polynomials -Points to remember

Important points to remember:

• A polynomial of degree 1 is a linear polynomial. e.g. 2x+3, 3x-2 etc
• A polynomial of degree 2 is a quadratic polynomial. e.g. 3x^2-3x+2 etc
• A monomial has 1 term. e.g.4 , 5x , 3x^2 etc
• A binomial has 2 terms. e.g. 2x+3 , 4x^-2 etc
• Degree of a polynomial is exponent of the highest degree term. e.g. degree of 3x+3 is 1.
• A real number k is called a zero of a polynomial P(x) if P(k) = 0.
• Graph of a linear polynomial is a straight line.
  Graph of a quadratic polynomial is a parabola.
• A polynomial of degree n has atmost n zeroes.
• Geometrically the zeroes of a polynomial are x coordinates of points where the graph of polynomial cuts/touches the x-axis.
• Relationship between zeroes and coefficients of a quadratic polynomial : In a quadratic polynomial ax^2+bx+c, a not equal to zero, a,b,c, are real numbers

Sum of zeroes = - coefficient of x/coefficient of x^2

Product of zeroes = constant term/coefficient of x^2

• Division Algorithm

Dividend = Divisor X Quotient + Remainder

MatheArt

MatheArt Kirigami is an art of Paper folding and cutting. Some interesting links to know more about it.

• http://lar.5u.com/kirigami.html


• http://www.flickr.com/photos/lilzabubba/sets/72157594397038216/


• Kirigami Applet http://members.aol.com/kevinsw/kweb/kirigami.html


• http://lydiard.free.fr/kirigami.htm

Curve Stiching

• http://www.montessoriworld.org/Handwork/stitch/stitch1.html

Verifying theoretical probability

Let us verify results obtained by theoretical approach of tossing a coin in probability.

For doing so we will use online applets for generating readings.

When we throw a coin, the probability of getting a head or a tail is 1/2.

Verify this result by actually throwing a coin 25 times.



Now, using the applet http://bcs.whfreeman.com/ips4e/cat_010/applets/Probability.html
Set the number of tosses = 50 , 100, 500 etc and note down the observations.

Note :If we increase the number of observations then we would get answers nearing to theoretical approach.

Birthday Problem

Dear students,
I read this statement about a famous birthday problem.In a group of 23 (or more) randomly chosen people, there is more than 50% probability that some pair of them will have the same birthday. For 57 or more people, the probability is more than 99%, tending toward 100%.

Do you want to try it out????
Click on the following link .You will get an applet through which you can generate birthdays randomly.

http://www-stat.stanford.edu/~susan/surprise/Birthday.html

Share your experience.

More on Probability...balls in a bag

Example 1 :There are 4 red balls and 3 green balls in a bag. One ball is taken out at random.
What is the probability of getting a red ball ?

Total no.of balls = 4+3 = 7
Favourable balls = 4 (as there are 4 red balls )

Required Probability = Total no. of fav. balls /Total no. of balls
= 4/7

What is the probability of not getting a red ball ?
Total no.of balls = 4+3 = 7
Favourable balls = 3 (as there are 3 balls other than red balls )

Required Probability = Total no. of fav. balls /Total no. of balls
= 3/7

Example 2 :There are 4 red balls , 5 blue balls and 3 green balls in a bag. One ball is taken out at random.
What is the probability of getting a red ball ?
Total no.of balls = 4+5+3 = 12
Favourable balls = 4 (as there are 4 red balls )

Required Probability = Total no. of fav. balls /Total no. of balls
= 4/12
=1/3
What is the probability of not getting a red ball ?
Total no.of balls = 4+5+3 = 12
Favourable balls = 8 (as there are 8 balls other than red balls )
Required Probability = Total no. of fav. balls /Total no. of balls
= 8/12
=2/3
What is the probability of not getting a red or a green ball ?
Total no.of balls = 4+5+3 = 12
Favourable balls = 9 (as there are 4 red balls and 5 green balls )
Required Probability = Total no. of fav. balls /Total no. of balls
= 9/12
=3/4

Probability is fun!

In my previous post I discussed about probability and some terms related to it. Today In the classroom I discussed the following important things .

• Why tossing of a coin is considered to be fair in deciding which team will have ball first in the game of football ?

This is because when we toss a coin we get a head or a tail and both these outcomes are equally likely.

• If a bag contain only red coloured balls ,then what is the probability of getting a green ball?

In this situation there is no favourable outcome as all the balls are red and we are asked to find the probability of getting a green ball. It is an impossible event. So, the required probability is zero.

• If a bag conatin only red coloured balls, then what is the probability of getting a red ball?

Every time we take out a ball, it will be red as there are only red coloured balls. This is a sure event. So, the required probability is 1.

Consider the following experiments...

Tossing a coin :When we toss a coin we get either a head or a tail . There are 2 possible outcomes. The probability of getting a head is 1/2 and probability of getting a tail is also 1/2.

Throwing a dice: When we throw a dice, we get 1,2,3,4,5 or 6.

Q A dice is thrown once, find the probability of the following:

a) getting an even number

b)getting a multiple of 3

c)getting a number greater than 3

d)getting a number greater or equal to 3

Solution a) Total outcomes = 6 {1,2,3,4,5,6}

Favourable Outcomes = 3 (as there are 3 even numbers i.e.2,4 or 6)

P(E) = Total no. of fav. outcomes / Total possible outcomes

So, P(getting an even number) = 3/6 = 1/2

b) Total outcomes = 6 {1,2,3,4,5,6}
Favourable Outcomes = 2 (as there are 2 multiples of 3 i.e. 3or 6)
P(E) = Total no. of fav. outcomes / Total possible outcomes
So, P(getting an even number) = 3/6 = 1/2

c) Total outcomes = 6
Favourable Outcomes = 3 (as there are 3 numbers greater than 3 i.e. 4,5or 6)
P(E) = Total no. of fav. outcomes / Total possible outcomes
So, P(getting an even number) = 3/6 = 1/2

d) Total outcomes = 6
Favourable Outcomes = 4 i.e. (3,4,5or 6)
P(E) = Total no. of fav. outcomes / Total possible outcomes
So, P(getting an even number) = 4/6 = 2/3

Pack of cards: Some facts

• Total cards = 52
• 2 colours : Red and Black
• Red cards = 26 ; Black cards = 26
• 4 suits : spade ,club, hearts and diamond
• 13 cards in each suit
• 3 face card in each suit : J ,Q,K are face cards
• Total 12 face cards
• 4 Kings, 4Quens and 4 Jacks
• Ace is not a face card

Probability

Today, we will learn about Probability. What is probability? It is the measure of uncertainity of an event. In our daily life we come across statements like Probably sheela will join tommorow, She may possibly come to school etc . In these statements there is uncertainity of happening of events. Now , some history about it... It started in 18th century in problems related to games of chance e.g. tossing a coin , throwing a die etc. What is an experiment? It means an activity which results in some outcomes. We will deal with Random experiments... It is that experiment which when repeat under same condition results in different outcomes. What is an event? The outcome of a random experiment is an event. e.g getting a head in tossing of a coin . Equally likely events... When we toss a coin , there is a fair chance of getting a head or a tail. These events are equally likely. Sample Space The collection of all possible outcomes of an experiment makes it sample space. We write a sample space as a set in which members are enclosed in curly brackets separated by comma. Consider an experiment Tossing a coin Sample Space is given by { H , T} I will talk about following experiments in my next post.

• Tossing a coin


• Throwing a dice


• Drawing a card from a pack of cards

Exploring Graphs using GeoGebra

You may learn and explore about linear equations in two variables by using tool GeoGebra

You may download interactive GeoGebra lessons from the following link

My GeoGebra Graph Lessons