We need to evaluate #color(red)((3x^2-8x+4)/(x-2))#
We will treat the given rational expression as follows:
Consider the Numerator first
#color(red)(3x^2-8x+4)# #.. color(red)(Expression.1)#
Evidently, this is a quadratic expression
We can factorize this quadratic expression:
To factor this quadratic expression, we will follow the procedure given below:
#color(green)(Step.1)#
We must split the coefficient of middle term into two numbers , such that when we add them we get the middle term, and when we multiply them we must get the product of the coefficient of the #x^2 term# and the constant,
Note that the product of the coefficient of the #x^2 term# and the constant is #(12)#,
#color(green)(Step.2)#
The two numbers are: #color(blue)(-6 and -2)#
When we add ( - 6) and ( -2 ) we get #(- 8)# and when we multiply the two values ( - 6) and ( -2 ) we get ( 3 X 4 = 12 )
Now, we write our #.. color(red)(Expression.1)# as follows:
#color(blue)(3x^2-2x - 6x+4)# #.. color(red)(Expression.2)#
#color(green)(Step.3)#
In this step, we break our #.. color(red)(Expression.2)# into groups:
#color(blue)(rArr (3x^2 - 2x) + (-6x +4))#
Factor out #color(green)(x)# from #color(blue)((3x^2 - 2x)# to obtain #color(blue)(rArr x*(3x - 2) )#
Factor out #color(green)(-2)# from #color(blue)((-6x +4)# to obtain #color(blue)(rArr -2*(3x -2) )#
#color(green)(Step.4)#
Using #color(green)(Step.3)# we can factor out the common term #color(blue)((3x-2)# and write the factors of our quadratic expression:
#color(blue)(rArr (3x -2) * (x - 2)#
#color(green)(Step.5)#
In this step we can rewrite our Numerator #color(red)(3x^2-8x+4)# as
#color(blue)(rArr (3x -2) * (x - 2)# #.. color(red)(Expression.3)#
#color(green)(Step.6)#
In this step, we will work on our given expression #color(red)((3x^2-8x+4)/(x-2))#
We can now write the above rational expression, using # color(red)(Expression.3)# as follows:
#color(blue)( {(3x -2) * (x - 2))/((x-2))}#
On simplification we get,
#color(blue)( {(3x -2) * cancel((x - 2)))/cancel((x-2))}#
Hence, #color(blue)(3x-2)# is our final answer.
I Hope this helps.
