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how can you define the following expresion in its smallest terms: 2/x-3 + 5x\(x-3)^2

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1 Answer

Hi, Bertu.

If I'm reading your question correctly, you are asking to find the lcm of the expression:
  2     +      5x                                                                  .
x - 3      (x - 3)^2

Do you see how the factor (x - 3) is duplicated?  I tell my students that when that happens, choose the "higher one" as your lcm (what I mean by "higher one" is the factor with the higher exponent value).  So the lcm of this expression is (x - 3)^2.

You didn't ask this, but just in case...

If the problem had asked you to simplify this expression, you would need to first rewrite it with equivalent fractions:

lcm is (x - 3)^2 ... so write both fractions with this denominator:

  2    •  (x - 3)        +       5x                                         .
x - 3 •  (x - 3)               (x - 3)^2

The new numerator for the first fraction is now 2x - 6:

   2x - 6         +        5x                                          .
(x - 3)^2              (x - 3)^2

Combine the numerators:

 2x - 6 + 5x                         .
  (x - 3)^2

Then combine like terms:

  7x - 6                                                    .
(x - 3)^2  

This is the original problem's expression simplified (just in case they ask)

Hope this helps you.