AI Tutor
When you reduce 1824\frac{18}{24} to 34\frac{3}{4}, you divide both numbers by their common factor of 66. But what happens when fractions are built out of variables and polynomials instead of plain numbers?
A rational expression is a fraction whose numerator and denominator are both polynomials.
This is formally known as 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.
Just as numerical fractions simplify by dividing out common factors (like 68=2⋅32⋅4=34\frac{6}{8} = \frac{2 \cdot 3}{2 \cdot 4} = \frac{3}{4}), rational expressions are simplified using 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.
How do we find these hidden matching pieces when polynomials are expanded?

The Factoring Method

To simplify any rational expression, you first factor completely both the top and bottom expressions into multiplied binomials or monomials. Once in factored form, you cancel any identical common factors shared by both parts.
Let's simplify x2βˆ’4x2+3x+2\frac{x^2 - 4}{x^2 + 3x + 2}. The numerator is a difference of squares (xβˆ’2)(x+2)(x - 2)(x + 2), while the denominator factors into (x+1)(x+2)(x + 1)(x + 2).
Dividing out the matching factor of (x+2)(x + 2) leaves our simplified answer: xβˆ’2x+1\frac{x - 2}{x + 1}.
A classic trap is trying to cross out individual added terms before factoring, like slashing the x2x^2 terms in x2βˆ’4x2+2\frac{x^2 - 4}{x^2 + 2}. You can only cancel terms that are multiplied, never terms that are glued together by addition or subtraction.
What happens when two binomials look almost identical, but their subtraction order is reversed?