# Simplifying polynomial fractions - Equality question

• Jul 24th 2012, 03:49 PM
ForumUser2
Simplifying polynomial fractions - Equality question
I am studying the simplification of polynomial fractions. My textbook has the example of:

\$\displaystyle \dfrac{2x+6}{x^2-9} = \dfrac{2}{x-3}\$ : factored out (x+3) from the numerator and denominator

My confusion is over the sense of 'equality' of the 2 expressions. I expected to be able to use either expression interchangeably as the definition for a function. However,

\$\displaystyle \ f(x) = \dfrac{2x+6}{x^2-9}\$ is undefined for x = { 3, -3 }

\$\displaystyle \ f(x) = \dfrac{2}{x-3}\$ is undefined for x = { 3 }

The domains of the functions differ. It appears that 'information' was lost in the simplification of the original expression.

Are the 2 expressions not really "equal"? Am I misunderstanding something about the concept of equality with regard to expressions?
• Jul 24th 2012, 04:14 PM
emakarov
Re: Simplifying polynomial fractions - Equality question
Quote:

Originally Posted by ForumUser2
Are the 2 expressions not really "equal"?

That's right. As numbers, \$\displaystyle \dfrac{2x+6}{x^2-9}\$ and \$\displaystyle \dfrac{2}{x-3}\$ are equal iff \$\displaystyle x\ne-3\$. To compare these expressions as functions, we must equip each of them with a domain. As long as those domains do not include -3, they are equal as functions (in the set-theoretic sense). With their natural, i.e., maximal, domains, they are different as functions because their domains are different.

In the process of transformations (e.g., when solving an equation), it is important to check if each equality is a true identity, i.e., holds for all values of variables. Similarly, it is important to check if two equalities or inequalities are truly equivalent, i.e., are both true or both false for all values of variables.
• Jul 24th 2012, 10:14 PM