SAT Math — rational expressions: common denominators and polynomial division

The Equivalent expressions skill of the digital SAT's Advanced Math domain includes rational expressions: not only cancelling a common factor, but combining fractions with different denominators, dividing one rational expression by another, and rewriting a quotient of polynomials in the form quotient plus remainder over divisor. This material trains those operations and the checks that catch errors.

You add and subtract rational expressions over a common denominator, with attention to the step where most errors happen — subtracting a whole binomial numerator — and one subtraction that simplifies to 1/(x + 3) after a factor cancels. You divide rational expressions by multiplying by the reciprocal and cancelling factors, simplify a complex fraction, and recognize a difference of two fractions with the same denominator that equals 1.

Polynomial division appears in several forms: dividing a quadratic by x + 2 and writing the remainder over the divisor, finding the constant A in 2x + 5 + A/(x − 1), dividing a difference of cubes, and an exact division by 2x + 1. A question finds a constant by matching numerators, and a context question writes the total time of a round trip at two speeds as one rational expression.

The material offers a quiz of twelve four-option questions, each with an explanation that shows the method and names the slip behind each wrong option, and a set of flashcards that collects the rules to remember. The explanations also show a quick numerical check — substituting a convenient value of x — that exposes a non-equivalent option, a habit that also works well with the graphing calculator available during the test.

The content is based on the College Board's published skill descriptions for the Advanced Math domain of the digital SAT Math section. It is independent practice and is not produced or endorsed by the College Board.

• Add and subtract rational expressions with unlike denominators
• Multiply and divide rational expressions and simplify by cancelling common factors
• Rewrite a polynomial division as quotient plus remainder over the divisor
• Simplify complex fractions
• Test equivalence of expressions by substituting a value

Practice material written by Zestly, based on the College Board's published description of the digital SAT Math section (Advanced Math domain: equivalent expressions — rational expressions and polynomial division), as given on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite.