My Best Mistake of the Week

This week, my best mistake happened when I tried to simplify a mixed-operations rational expression. I understood how to add fractions, but when division and factoring were involved, I forgot the correct order of operations and didn’t factor before simplifying.

The question I made was:

What I Learned

From this mistake, I learned the correct steps for simplifying mixed rational expressions:

• Rewrite division as multiplication by the reciprocal
• Factor every polynomial before simplifying
• Cancel common factors
• Create a common denominator only when adding
• Rewrite the final expression properly

Steps That I Did

1. Rewrite the division as multiplication
   
2. Factor all expressions
   
   
   So it becomes:
   
3. Cancel common factors
   Cancel and :
   
4. Substitute this back into the original expression
   Now the whole question becomes:
   
5. Factor where possible
   
   
6. Add the fractions
   

Restrictions

Reflection

This mistake helped me understand the importance of factoring first, flipping the second fraction when dividing, canceling carefully, and only adding after rewriting fractions with a common denominator. Once I did that, the expression made sense and became much easier to solve.