We need to evaluate color(red)((3x^2-8x+4)/(x-2))3x2−8x+4x−2
We will treat the given rational expression as follows:
Consider the Numerator first
color(red)(3x^2-8x+4)3x2−8x+4 .. color(red)(Expression.1)..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)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 termx2term and the constant,
Note that the product of the coefficient of the x^2 termx2term and the constant is (12)(12),
color(green)(Step.2)Step.2
The two numbers are: color(blue)(-6 and -2)−6and−2
When we add ( - 6) and ( -2 ) we get (- 8)(−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)..Expression.1 as follows:
color(blue)(3x^2-2x - 6x+4)3x2−2x−6x+4 .. color(red)(Expression.2)..Expression.2
color(green)(Step.3)Step.3
In this step, we break our .. color(red)(Expression.2)..Expression.2 into groups:
color(blue)(rArr (3x^2 - 2x) + (-6x +4))⇒(3x2−2x)+(−6x+4)
Factor out color(green)(x)x from color(blue)((3x^2 - 2x)(3x2−2x) to obtain color(blue)(rArr x*(3x - 2) )⇒x⋅(3x−2)
Factor out color(green)(-2)−2 from color(blue)((-6x +4)(−6x+4) to obtain color(blue)(rArr -2*(3x -2) )⇒−2⋅(3x−2)
color(green)(Step.4)Step.4
Using color(green)(Step.3)Step.3 we can factor out the common term color(blue)((3x-2)(3x−2) and write the factors of our quadratic expression:
color(blue)(rArr (3x -2) * (x - 2)⇒(3x−2)⋅(x−2)
color(green)(Step.5)Step.5
In this step we can rewrite our Numerator color(red)(3x^2-8x+4)3x2−8x+4 as
color(blue)(rArr (3x -2) * (x - 2)⇒(3x−2)⋅(x−2) .. color(red)(Expression.3)..Expression.3
color(green)(Step.6)Step.6
In this step, we will work on our given expression color(red)((3x^2-8x+4)/(x-2))3x2−8x+4x−2
We can now write the above rational expression, using color(red)(Expression.3)Expression.3 as follows:
color(blue)( {(3x -2) * (x - 2))/((x-2))}(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.
