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If you saw the fraction 610\frac{6}{10}, you would instantly spot the shared factor of 22 and reduce it to 35\frac{3}{5}. But what happens when the numbers are replaced by full quadratic polynomials?
A rational expression is simply any fraction where the numerator and denominator are polynomials. Just like numeric fractions, we simplify them by finding and dividing out common factors β€” quantities that are multiplied together, never individual added terms.
This relies on the Fundamental Principle of Rational Expressions: A⋅CB⋅C=AB\frac{A \cdot C}{B \cdot C} = \frac{A}{B} (where B,C≠0B, C \neq 0). Under this principle, you may only divide out and cancel identical, non-zero multiplicative factors.
This rule is the Fundamental Principle of Rational Expressions: A⋅CB⋅C=AB\frac{A \cdot C}{B \cdot C} = \frac{A}{B} (where B,C≠0B, C \neq 0). It guarantees that dividing out identical, non-zero multiplicative factors leaves an equivalent fraction.
How do we find those hidden common factors when expressions look completely different at first glance?

Factoring Numerators and Denominators

To simplify a rational expression, completely factor both the top and bottom into linear terms using techniques like greatest common factors, differences of squares, and quadratic trinomial factoring. Once factored, any identical factor pair in the numerator and denominator divides to 11.
A common trap is attempting to cancel terms before factoring, such as crossing out x2x^2 in x2βˆ’4x2+2\frac{x^2 - 4}{x^2 + 2}. Addition binds numbers together into an unbreakable unit β€” unless the entire grouping (x2βˆ’4)(x^2 - 4) matches the bottom exactly, you cannot divide parts of it.
What happens when two binomial factors have the exact same terms, but their subtraction is flipped backward?

The Opposite Factor Property

When you encounter reversed terms like (aβˆ’b)(a - b) and (bβˆ’a)(b - a), notice that factoring out a βˆ’1-1 from (aβˆ’b)(a - b) produces βˆ’1(βˆ’a+b)-1(-a + b), which is simply βˆ’1(bβˆ’a)-1(b - a). Therefore, any ratio of opposite factors aβˆ’bbβˆ’a\frac{a - b}{b - a} simplifies directly to βˆ’1-1.