Key takeaway: A rational expression is a fraction whose numerator and denominator are polynomials, and it is undefined for any value of the variable that makes the denominator zero.
Grade 11 – Algebra 2A-APR.D.6Expressions and Equations
Definition of Rational Expression
A rational expression is a quotient of two polynomials, in the same way that a rational number is a quotient of two integers. The expression \(\frac{x+4}{x+3}\) qualifies, and so does \(\frac{7}{x^2-1}\), because the top and the bottom are both polynomials.
Working with a rational expression feels like working with fractions: you factor, cancel common factors, then multiply, divide, add or subtract. One extra duty applies. Any value of the variable that makes the bottom equal to zero is not allowed, and you must state it.
Formula and notation
In general a rational expression looks like
\[\frac{P(x)}{Q(x)}, \quad Q(x) \neq 0,\]
where \(P\) and \(Q\) are polynomials. The condition \(Q(x)\neq 0\) lists the excluded values, which you find by solving \(Q(x)=0\) before any cancelling.
Key facts
- Only common factors cancel; terms joined by plus or minus signs never cancel.
- Excluded values come from the original denominator, even after a factor cancels.
- Multiplying fractions of this type means factor first, then cancel across.
- Adding two of them needs a common denominator, found from the factored forms.
Rational Expression example
Simplify \(\frac{x^2-16}{x^2-x-12}\) and list every value of \(x\) that must be excluded.
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Factor the top with a difference of squares: \(x^2-16=(x-4)(x+4)\).
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Factor the bottom: \(x^2-x-12=(x-4)(x+3)\).
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Cancel the shared factor \(x-4\) to get \(\frac{x+4}{x+3}\).
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The original bottom is zero at \(x=4\) and \(x=-3\), so both stay excluded.
The result is \(\frac{x+4}{x+3}\) with \(x\neq 4\) and \(x\neq -3\).
Common mistake
Watch out
Tavian simplifies \(\frac{x+7}{x+2}\) by crossing out both x’s and writes \(\frac{7}{2}\).
Why it is wrong
The letter x is a term in a sum, not a factor of the whole top or bottom. Test \(x=1\): the expression equals \(\frac{8}{3}\), not \(\frac{7}{2}\). Only factors that multiply the whole numerator and the whole denominator can cancel.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Expressions and Equations