< > Inequalities ≤ ≥

This subtopic reminds me a lot of how my add math teacher, taught me doing the inequalities in the quadratic function. I used to forget and mix up with the sign changing of the inequality sign when negative turns positive and vice versa. Sometimes, I even felt Advanced Functions is a continuous of my secondary school add math class and it is also a subject that brings back a lot of beautiful memories of mine when I was in secondary school. But, one thing I prefer Advanced Functions is that it is way more interesting than add math. last time i used to do piles and piles of homework and practices. But, now things changed. We do mind maps and online activities. It's fun!!! That's why I'm doing my homework now... actually I should say blogging... hahaha! 


OK! Enough for my the rubbish-like introduction take make a little sense but not too much sense... OMG... what I'm saying!  


In this subtopic called "Solve Inequalities Using Technologies" I found out, we don't really use much technology like graphing calculator. But, we use different ways like number line method, graphing method and numeric method and 2 graph method. 


First, the number line method.
For inequality sign "<" the arrow on the number line goes towards left as it is less than certain number.
For inequality sign ">" the arrow on the number line goes towards right as it is greater than certain number. 


Watch this video about Graphing Inequalities on a Number Line.












From the example above we noticed one thing. A fully colored dot and a hollow dot. What is the difference?
a fully colored dot on the number lines mean the sign  "≤" or "≥" depending on the direction of the arrow. A hollow dot means the sign "<" and ">" depending on the direction of the arrow. 


Second, the graphing method or the 2 graph method.


By looking at the graph we can know which part of the graph is above or below the x-axis. This method provides us an image of the range where the y-values are positive or negative and thus giving us the answer of the inequalities. 


If we solve the inequalities using 2 graphs method, we need to make both side of the inequalities to be equal to y then plot 2 graphs. The x-value of the intersection point for both graphs will be the answer for the inequalities.


For a clearer understanding of graphing method, I suggest you to go to this link. http://www.sosmath.com/algebra/inequalities/ineq04/ineq04.html. I think it is pretty informative and clear.


For the last method, Numerical method, I found it the best method. I do not know why, i just like it the most. That's why people says "love is blind"... haha!


First we transform the inequalities into an equation that is equal to 0. Then, we look for its x-intercept. Then we test each interval of the x-intercept. 


Let's say for example the x-intercept are -6 and 2, we can try out when x<-6, the function is either positive or negative. Then the interval between -6<x<2 and x>2. 


Then we look for the interval that is positive or negative, according to the sign of the inequalities in the question, and that interval will be our answer. For example, for question "solve x³+2x-1>0" we will look for the positive part of the function as the sign is greater than and vice versa for question with a less than sign.



Web resources: 


Youtube, Graphing Inequalities on a Number Line, <http://www.youtube.com/watch?v=AjZzACaeIco&feature=player_embedded>


S.O.S Math, Solving Polynomial Inequalities by Graphing, <http://www.sosmath.com/algebra/inequalities/ineq04/ineq04.html>

What?! Polynomial Functions has its Family?

In a family, we have some similarities with our parents and siblings and other relatives. It can be the physical appearance or the characteristic. Same goes to the Polynomial equation of a family! I know, you might think what I am talking about absolute ridiculous rubbish. Hahaha~ chill~


First of all, let me explain what is the families of polynomial functions. A polynomial function that belongs to a family have the same characteristic. Polynomial function of the same family have the same x-intercept, roots, and zeros.


Example:


f(x)=(x+1) (x-1) (x+3) and
f(x)=2(x+1) (x-1) (x+3) 
belong to the same family because they have the same x-intercepts, roots and zeros. 


Example 2: 

Three polynomial equations above are of the same family because they have the same x-intercept of 2 and 5.


To determine the equation of the particular member of the family, the only thing that we require is any one point on the graph. We can substitute the x value or y-value into the function and find out the variable that differs the shape of the graph.


In general, polynomial functions of the same family can be represented in the form, 


y = k( x - a₁ ) ( x - a₂ )...( x - an ), where k ∈ R, k ≠ 0.

Birds of a feather flock together. 
Polynomial equation of a family, share the same x-intercept together. 




See? I told you polynomial functions have their own family. Haha! Enjoy~


Web resource:The Family of Quadratic Functions <http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_08/other_calculator_keystrokes/472_Alg1_9-1B_873823_CFX.pdf>

Zer0, Roots and x-intercept. What's Their Relationship???

After I learned the Chapter 2.3, the question of "what's the difference and relationship between zero, root, and x-intercept?" pop in my mind. Why do they have different names yet they are almost the same????? Weird right? :P


Then. I found out they are actually interrelated and they are same in terms of its value.
Let's say for example the graph below.
 
The x-intercept on the graph is the zero of the graph and the polynomial equation; the zero of the graph is the factor of the polynomial equation; the x-intercept and zero are the roots of the polynomial equation. Thus, the zero, roots and x-intercepts are -1 and 5 (as from the graph above). 


BUT, this is only for those polynomial equation that is factorable. For those polynomial equation that are not factorable, we can find out their zero or roots or x-intercept (if they have roots), we need to use the quadratic formula 

or by using our graphing calculator.


Therefore, as a conclusion:

• the real roots of a polynomial equation P(x)=0 is similar to the x-intercept of the graph P(x) and vice versa.
• A factorable polynomial equation's roots are equal to its zero and thus solving the factor of it.
• BUT, when we come to polynomial equation that is not factorable, we need to solve it using technology or by using graphing method. 

Web resources: Wikipedia, "Zero of a Function" <http://en.wikipedia.org/wiki/Zero_of_a_function>

                        
The Math Page, "The Roots, or Zeros, of a Polynomial"          <http://www.themathpage.com/aprecalc/roots-zeros-polynomial.htm>

                       

The Factor Theorem

When I learn about Factor Theorem, I found four things that I think are the essence of the theorem.

First, the factor theorem itself. A brief and quick summary for it:

x-b is a factor of a polynomial P(x) if and only if P(b)=0.

Similarly, ax-b is a factor of P(x) if and only if P(b/a)=0.

Note: the acronym I learnt in the class is "iff" short form of if and only if. So don't be shock if      you see your Advanced Function teacher write iff! Definitely not spelling error. =D

Example:

x-1 is the factor of P(x)= x³ + 4x² + x - 6 because P(1) = 0

x+1 is not the factor of P(x)= x³ + 4x² + x - 6 because P(-1) = 0

Second, the integral zero theorem. A brief explanation of this theorem:

If x-b is a factor of a polynomial function P(x) with leading coefficient 1 and remaining coefficient that are integers, then b is a factor of the constant term of P(x). 

Thus, as a summary of this theorem, the word zero means the value of b when polynomial function P(x)=0.

Example:

P(x)= x³ + 2x² - 5x - 6

According to the Integral Zero Theorem, the possible factors of the constant term of P(x) MIGHT BE the possible factor in the form of x-b of P(x).

So, the zeroes of the constant term are: ±1, ±2, ±3 and ±6. After that, substitute all those zeroes into P(x). In this case, the zeroes that make P(x)=0 are -3, -1 and 2. Thus, the factors are x+3, x+1, and x-2.

Third, the Rational Zero Theorem. A brief explanation on it:

P(x) is a polynomial function with all integers coefficient and a leading coefficient that is greater than 1 or not an integer, x=b/ is the zero of P(x) where a and b are integers. Then, 

• b is the factor of the constant term,
• a is the factor of the leading coefficient,
• ax-b is the factor of P(x).

Example:

The polynomial function of P(x)= 3x³ + 2x² - 7x + 2 needs Rational Zero Theorem to solve it. 

The factors of the leading coefficient are ±1 and ±3. The factors of the constant term are ±1 and ±2. Thus, the possible combination of zeroes that can make P(b/a) are ±1/1, ±1/2, ±3/1 and ±3/2. In this case, the zeroes that make P(b/a)=0 are 1, -2 and 1/3. Thus, the factors of P(x) are x-1, x+2 and 3x-1.

Fourth, the grouping factorization method. I found this method is quite convenient but it is different from what we (Malaysia national education system's students) learn in form 4 and 5. Don't freak out, you will find it easy after you read the following.


Grouping factorization method can be applied when pairs of term of polynomial can be grouped to factor out a common factor so that the resulting binomial factors are the same.


Example:


f(x)= x³ + 2x² - 9x -18.                 <~ group out the binomial factors
     = x² (x + 2) - 9 (x + 2)             <~ group the remaining numbers
     = (x + 2) ( x² - 9 )                    <~ factorize the factor that has power that is great than 1
     = (x + 2) (x + 3) (x - 3 )           <~ DONE ^^ 

That's it for the Factor Theorem. If you worry I make you blur (definitely I hope I'm not!!!), just youtube-ing the video below. I found it very useful. Thanks for reading and "digesting" what I posted! 

Web resources: Youtube, "The Factor Theorem, ExamSolution", <http://www.youtube.com/watch?v=6G3iAgpK4kA&feature=player_embedded>


Purplemath, "The Factor Theorem", <http://www.purplemath.com/modules/factrthm.htm>

A cute little song to relax~

Hello people, I found a cute little song. I like the last line that says, "Yes try as you may,you just can't get away FROM MATHEMATICS." Personally, I am the type of person who can't get away of maths if I could not get it solved! Is it quite sad to hear. But, no pain, no gain. The more I suffer as a result I can't get it away, the more the satisfaction I got from it once I get it solved! I think all maths freak are the same! gimme 5! piakkkk! XD 


Web resource: Youtube, "Math song", <http://www.youtube.com/watch?v=ZPMRA4yFeeM>

The Remainder Theorem

REMAINDER THEOREM is a way for us to solve polynomial equations in order to get us the remainder or to check its factor.


Example,
P(x) = 2x³ - 2x² - 3x + 3


This function can be either solved by using long division with factor of x - b where b is a real number or solve by using Remainder Theorem by substituting the value of b into P(x) where x = b from the factor of x - b into P(x). For example, when x - 2 is the factor of P(x),  then x = 2.  Then, substituting x = 2 into P(x). 


P(x) = 2x³ - 2x² - 3x + 3
P(2) = 2(2)³ - 2(2)² - 3(2) + 3
P(2) = -1
Thus, the remainder is -1.


Thus, the Remainder Theorem:


when a polynomial function of P(x) is divided by x-b, the remainder is P(b). where b is an integers.


iExmaple 2:


P(x) = 2x³ - 2x² - 3x + 3, divisor, 2x-3


When the divisor has a leading coefficient that is not 1, then we assume the leading coefficient as "a" and the constant as "b". Thus, x=b/a.


P(x) = 2x³ - 2x² - 3x + 3
P(3/2) = 2(3/2)³ - 2(3/2)² - 3(3/2) + 3
P(3/2) = Remainder


Thus, the Remainder Theorem:


when a polynomial function of P(x) is divided by ax-b, the remainder is P(b/a). where b and a are  integers.


There are two things that we need to remember. To write the result we got after doing long division or solving using remainder theorem, there are 2 ways. It can be either in quotient form or statement form (can used to check the division)


Let Q(x) be the quotient,
R be the remainder;
P(x) be the polynomial function;
x-b be the divisor


quotient form : P(x) = Q(x) + R 
                          x-b                x-b


statement : P(x)=  [Q(x)] [(x-b)] + R


If we wanna explain the remainder geometrically, the quotient is actually the slope of secant between point (b, P(b)) and (x, P(x))


to get a better concept of Remainder Theorem, I found you all a video.



This is an easy theory, so, just make use of it well. enjoy~ Thanks for reading!


Web resources: Youtube, "The Remainder Theorem, Exmaple 2", <http://www.youtube.com/watch?v=q57NU29KNGw>


Purplemath, "The Remainder Theorem", <http://www.purplemath.com/modules/remaindr.htm>

Before we start...

The first day we started Chapter 2, we learned about long division which Ms. Joanne (our Advanced Functions teacher) call it "Long way". This "long way" makes everyone in my class burst of laughing because we have a friend in the class called, Long Wei! It's the same pronunciation. Sorry Long Wei, no offense~

Back to the topic, long division is the prerequisite skill for us to learn remainder theorem and factor theorem. We will need it very much in solving polynomial equation in this chapter. 

This might look childish and you take it very easy as you learn it since you were in primary school. But, never underestimate it. It is so important throughout this chapter. Haha!


A brief description:
68 is the dividend;
2 is the divisor;
34 is the quotient;
and 8 is the remainder.


When you are doing long division, you must be very careful. A small careless mistake might get you a wrong answer. So, we must times the divisor and quotient meticulously, and minus the answer carefully.


For more information, go to this link, http://en.wikipedia.org/wiki/Long_division. I hope it helps you!


Web resources: Wikipedia, "Long Division", <http://en.wikipedia.org/wiki/Long_division>


Purplemath, "Polynomial Long Division", <http://www.purplemath.com/modules/polydiv2.htm>

~The Grand Opening~

Hi, everyone! This is my first ever blog and i give it to my CPU's Advanced Function. I hope I do it well. XD Basically, this is a blog for everyone to discuss about advanced functions. In this blog, I'll emphasize on the Unit 2 of the Advanced Function from what I learned from Ms. Joanne, my dear teacher. I do not want the blog to be too dull and academic-like. Instead, I hope it is a great place for everyone to come and comment and get some handy tips or information about Advanced Function.


Advanced Function isn't something dry and boring and scary and just numberS! So, enjoy! HANG ON! IT'S ADVANCED FUNCTION lah~ ^^