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>