ACT Mathematics — factoring, rational, radical, exponential and absolute-value equations

ACT describes its Algebra subcategory as solving, graphing and modeling many kinds of expressions and equations, including linear, polynomial, radical and exponential relationships. This material covers the non-linear side of that list, the algebra that separates a quick, confident answer from a long detour.

It starts with factoring patterns: a common factor followed by a difference of squares, a difference of cubes, and a four-term polynomial factored by grouping. A rational expression is then simplified by cancelling a common factor, never a common term. The equation questions each carry a classic trap. A rational equation produces a single candidate that makes a denominator zero, so it has no solution at all. A radical equation produces two candidates after squaring, and only one survives the check in the original equation. An exponential equation is solved by rewriting both sides with the base 2, and 2^x = 20 by taking logarithms, since 20 is not a power of 2.

Absolute value appears as an equation with two cases and as an inequality that describes a single interval centered at 4. The material closes with the sums of an arithmetic and a geometric sequence, with the sum formulas given in the question as ACT does for formulas beyond the basics: the task is to find the right last term and to substitute correctly.

The material offers two formats. The quiz has twelve multiple-choice questions with four answer choices each, as on the enhanced ACT, and every explanation names the slip behind each wrong choice, so a miss tells you what to fix. The flashcards collect the rules and formulas worth knowing by heart.

The content is based on ACT's published description of the mathematics section (reporting categories and the topics listed for each). It is independent practice and is not produced or endorsed by ACT.

• Factor polynomials using common factors, special products and grouping
• Simplify rational expressions
• Solve rational and radical equations and reject extraneous solutions
• Solve exponential equations with a common base or with logarithms
• Solve absolute-value equations and inequalities and find sums of arithmetic and geometric sequences

Practice material written by Zestly, based on ACT's published description of the ACT mathematics section (act.org, Mathematics Test Description: Preparing for Higher Math — Algebra). Formulas beyond the basics are given in the questions, as ACT states that recall of complex formulas is not required.