Quick Answer
A rational expression is a fraction where both the numerator and denominator are polynomials. On the Digital SAT, these concepts typically appear in the Passport to Advanced Math section. Students frequently encounter these in approximately 2 to 4 questions per test, often requiring simplification or finding excluded values for the variable.
A rational expression is an algebraic fraction of the form P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) is not equal to zero. It represents a ratio of two polynomial functions and is undefined at any x-value that results in a zero denominator.
Simplify the expression: (x^2 - 9) / (x^2 + 5x + 6). Solution: Factor both parts: [(x - 3)(x + 3)] / [(x + 2)(x + 3)]. Cancel the common factor (x + 3). The simplified expression is (x - 3) / (x + 2), where x is not equal to -2 or -3.
Canceling terms across addition: Students often incorrectly cancel individual terms like 'x' in (x+5)/x, forgetting that only factors of the entire numerator and denominator can be canceled.
Ignoring domain restrictions: Forgetting that an expression is undefined when the denominator is zero, even if the factor causing that zero is canceled out during the simplification process.
Incorrect sign distribution: Failing to distribute a negative sign to every term in the numerator when subtracting two rational expressions with a common denominator.
Students targeting 750+ should know that the SAT often masks rational expression problems as 'equivalent expression' questions. If factoring seems impossible, you can sometimes substitute a small, easy integer for the variable in both the original expression and the answer choices to find a numerical match, provided the number does not make any denominator zero.
Asymptote
An asymptote is a line that a graph approaches but typically never touches as it extends toward infinity. On the Digital SAT, asymptotes appear in the Advanced Math section, primarily within rational and exponential function questions, occurring in approximately 1–2 problems per test to assess end-behavior analysis.
Expression
An expression is a mathematical phrase combining numbers, variables, and operators without an equals sign. On the Digital SAT, expressions are fundamental to the Algebra and Advanced Math sections. Typically, approximately 20-30% of Math questions involve manipulating or simplifying expressions, appearing frequently in both multiple-choice and student-produced response formats.
Factoring
Factoring is the mathematical process of breaking down a polynomial into a product of simpler expressions or factors. On the Digital SAT, this technique is frequently tested in the Math modules, appearing in approximately 10-15% of algebra and advanced math questions, often requiring students to identify equivalent expressions or find the zeros of quadratic functions.
Polynomial
A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents. On the Digital SAT, polynomials frequently appear in the Advanced Math section, typically requiring students to add, subtract, multiply, or factor expressions. These questions often represent approximately 10-15% of the math content across both modules.
Rational Equation
A rational equation is an algebraic equation containing at least one fraction with a variable in the denominator. On the Digital SAT, these typically appear in the Math section as medium-to-hard difficulty problems. Students encounter them approximately 2-4 times per test, often requiring solving for a specific variable or identifying extraneous solutions.
A rational expression on the SAT is a mathematical expression written as a fraction where both the numerator and denominator are polynomials. It is a core component of the Passport to Advanced Math section. These expressions represent ratios and are used to test a student's ability to factor polynomials, simplify complex algebraic structures, and understand function properties where the denominator cannot be zero.
To simplify a rational expression, you must first factor both the numerator and the denominator completely into their prime polynomial factors. Once factored, identify any common factors present in both the top and bottom. By dividing both the numerator and denominator by these common factors—effectively 'canceling' them—you reduce the expression to its simplest form while maintaining the original domain restrictions.
The difference between a rational expression and a rational equation is that an expression is a single mathematical phrase without an equals sign, while an equation sets two expressions equal to each other. You simplify or rewrite an expression to find an equivalent form, but you solve an equation to find the specific value of a variable that makes the statement true.
On the Digital SAT, you will typically encounter approximately 2 to 4 questions that directly involve rational expressions or their simplification. However, the underlying skills, such as factoring quadratics and polynomial arithmetic, are foundational for many other question types in the Math modules. Proficiency in this area is essential for scoring high in the Passport to Advanced Math category.
