Introduction
Rational expressions — fractions whose numerator and/or denominator is a polynomial — appear regularly in the Advanced Math domain, which makes up roughly 35% of the SAT Math section. Questions ask you to rewrite a rational expression in an equivalent form: cancel common factors, combine fractions over a common denominator, or carry out polynomial multiplication before simplifying. Mastering this skill also unlocks equation-solving questions where both sides are rational expressions.
Core Concept
A rational expression has the form , where and are polynomials and .
The golden rule: a rational expression is only defined where its denominator is nonzero. Any -value that makes any denominator equal zero across the entire simplification process must be excluded from the domain, even if that factor cancels out. Those exclusions create holes in the graph of the expression.
The four operations follow the same rules as numeric fractions:
| Operation | Rule |
|---|---|
| Simplify | Factor top and bottom; cancel common factors. |
| Multiply | Multiply numerators; multiply denominators; then simplify. |
| Divide | Multiply by the reciprocal of the divisor; then simplify. |
| Add / Subtract | Rewrite each fraction over the LCD; combine numerators. |
Quick illustration — why factoring comes first:
The factor cancels, but the restriction stays because the original expression was undefined there. See the figure below for a visual of how the cancelled factor creates a hole.
Unlock the full Equivalent Expressions note with Nova
You're reading the preview. Unlock the complete note — every worked example, examiner pitfall and practice question — plus 24/7 AI tutoring from Nova that teaches directly from these notes.