AI Tutor
If you look at the fraction x+4x+8\frac{x + 4}{x + 8}, is it tempting to cross out the xx's and get 48=12\frac{4}{8} = \frac{1}{2}?
Try plugging in x=4x = 4: you get 4+44+8=812=23\frac{4 + 4}{4 + 8} = \frac{8}{12} = \frac{2}{3}, which is definitely not 12\frac{1}{2}! Let's see why that happens and how to simplify these expressions correctly.

Factors vs. Terms

A rational expression is simply a fraction where the numerator and denominator are polynomials.
You can only cancel common multiplicative factors, never individual additive terms. This cancellation relies on the Fundamental Principle of Rational Expressions: a⋅cb⋅c=ab\frac{a \cdot c}{b \cdot c} = \frac{a}{b} (where b≠0b \neq 0 and c≠0c \neq 0).
This canceling relies on the Fundamental Principle of Rational Expressions: a⋅cb⋅c=ab\frac{a \cdot c}{b \cdot c} = \frac{a}{b} (where b≠0b \neq 0 and c≠0c \neq 0), because cc=1\frac{c}{c} = 1.
So before you can cancel anything in a polynomial fraction, how do you turn added terms into multiplied factors?

The Factoring Strategy

To simplify a rational expression, fully factor both the top and bottom polynomials first, then cancel every common factor shared by both.
Let's simplify x2βˆ’9x2+5x+6\frac{x^2 - 9}{x^2 + 5x + 6}. First, factor the top as a difference of squares (xβˆ’3)(x+3)(x - 3)(x + 3), and factor the bottom into two binomials (x+2)(x+3)(x + 2)(x + 3).
Since (x+3)(x + 3) appears in both numerator and denominator, dividing them out leaves the simplified result: xβˆ’3x+2\frac{x - 3}{x + 2}.