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>