SAT Math Strategy

Rational Expressions on the SAT

Simplify with Domain Awareness

Build equations from context, spot patterns fast, and practice with intent.

5 Min Read
Math Skill
Equation-First
Practice Qs

Why the SAT Emphasizes Rational Expressions

Rational expressions are fractions made from polynomials, and the SAT uses them to test whether you can simplify correctly and keep track of restrictions. The algebra looks simple, but the logic matters.

This lesson shows how to factor first, cancel only common factors, and state excluded values clearly. You will also see why canceling terms instead of factors creates incorrect results.

A Simple Definition Unlocks Rational Expressions

Treat a rational expression like a normal fraction. You can only cancel factors, not terms. That means you must factor both the numerator and the denominator before you simplify.

Every value that makes the original denominator zero must be excluded from the domain, even if the factor cancels later. Those excluded values create holes in the graph and can change the meaning of the expression.

Work Through Rational Expressions Step by Step

Guiding Question

Simplify $\\frac{x^2 - 9}{x^2 - 3x}$.

Simplify the expression carefully and state the excluded values so the domain is correct.

Start with the expression in its original form.

\frac{x^2 - 9}{x^2 - 3x}

Factor both the numerator and denominator so you can cancel safely.

\frac{(x - 3)(x + 3)}{x(x - 3)}

Cancel the common factor to simplify the rational expression.

\frac{x + 3}{x}

State the excluded values that make the denominator zero.

x \neq 0,\; x \neq 3

Use Desmos to Check Rational Expressions

Guiding Question

Simplify $\\frac{x^2 - 9}{x^2 - 3x}$.

Desmos can help you spot holes and compare shapes, but algebra determines the excluded values. Use Desmos to check a few values or see whether graphs disagree, then rely on algebra for the exact simplification.

Graph the original expression so you can compare its shape and any holes.
Desmos y = (x^2 - 9) / (x^2 - 3x)
Graph the simplified expression to see if it matches where both are defined.
Desmos y = (x + 3) / x

Algebra is faster for exact simplification and excluded values. Desmos is helpful for checking or spotting holes after you simplify.

Expert move: Use Desmos to disprove an answer quickly - if values or graphs differ at any $x$, the expressions are not equivalent - then use algebra to confirm the exact simplification.

  • Desmos features used: graphing, overlay comparison.
  • Common mistake: canceling terms instead of factors.

Practice Rational Expressions with SAT-Style Questions

Simplify and track excluded values.

medium

Simplify \frac{x^2 - 16}{x^2 - 4x} .

easy

Which values are excluded from \frac{1}{x^2 - 9} ?

medium

Simplify \frac{x^2 + 5x + 6}{x^2 + 2x} .

easy

For \frac{2}{x - 5} + \frac{1}{x - 5} , what is the simplified result?

Key Takeaways to Remember for Rational Expressions

  • Factor first, then cancel factors, not terms.
  • Always state excluded values from the original denominator.
  • Desmos can confirm equivalence and reveal holes.