Simplifying Expressions: Understanding Solutions and Common Factors

[Bardagath]

It is a case of simplifying this expression:

a^2 - ab / ab



The solution given in a textbook is:

a - b / b



I observe that for the one "a" that was canceled below, two "a's" were canceled above in this simplification.


Why would the solution not be: a^2 - b / b? I would like to know what happened to the a in a^2 if the textbook answer is the correct answer.


I am also having problems simplifying this expression: 3y^2 - 27 / 12y^2 + 36y
I have tried factorizing to find common factors that cancel each other out but have not had any success... I know what the solution is but I would like to know how to get there.

 

[HallsofIvy]

No, only one a was canceled in both numerator and denominator. Remember that you can only cancel things that are multiplied. You are thinking
"Okay, I will cancel the last a in a²- ab with the a in the denominator and get (a²-b)/b."

but you can't do that: the second a is not a factor. What you need to do is first factor a²- ab= a(a- b). NOW you can cancel:
\frac{a^2- ab}{ab}= \frac{a(a-b)}{ab}= \frac{a- b}{b}
where you have canceled the a multiplying (a- b) in the numerator with the a multiplying b in the denominator.

 

[tiny-tim]

Welcome to PF!

(a^2 - ab )/b

Hi Bardagath! Welcome to PF!

I suspect you're having vision problems … you can't take in the whole top line in one go.

Just split it into two fractions, then factor it …

(a^2)/b - ab/b = … ?

I am also having problems simplifying this expression: (3y^2 - 27)/(12y^2 + 36y)
I have tried factorizing to find common factors that cancel each other out but have not had any success... I know what the solution is but I would like to know how to get there.

With ordinary fractions (just numbers), if asked to simplify 21/4, you might say 5\frac{1}{4}\,.

5 is the "whole" multiple, and 1 is the remainder.

With polynomial fractions, you do the same … for example, (3x + 11)/(x + 2) = 3 + 5/(x + 2).

3 is the "whole" multiple, and 5 is the remainder.

The remainder can be a number, as 5 above, or it can be a polynomial (of lesser degree than the denominator. of course!)