Simplifying Expressions: Understanding Solutions and Common Factors

  • Thread starter Bardagath
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In summary, when simplifying polynomial fractions, you can use long division or factoring to find the "whole" multiple and the remainder, just like with ordinary fractions. Then, you can rewrite the fraction as the whole multiple plus the remainder over the denominator.
  • #1
Bardagath
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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.
 
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  • #2
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:
[tex]\frac{a^2- ab}{ab}= \frac{a(a-b)}{ab}= \frac{a- b}{b}[/tex]
where you have canceled the a multiplying (a- b) in the numerator with the a multiplying b in the denominator.
 
  • #3
Welcome to PF!

Bardagath said:
(a^2 - ab )/b

Hi Bardagath! Welcome to PF! :smile:

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 = … ? :smile:
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 [tex]5\frac{1}{4}\,.[/tex]

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!) :smile:
 

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