Algebra

Determine all applicable classifications of a real number as natural, integer, rational, or irrational.

Evaluate algebraic expressions by substituting specified values for variables.

Translate verbal relationships into algebraic expressions.

Rewrite expressions using the commutative, associative, and distributive properties.

Identify and use additive and multiplicative identities and inverses.

Simplify linear expressions by combining like terms and justify each step with an algebraic property.

Explain why adding the same real number to both sides, or multiplying both sides by the same nonzero real number, preserves the solution set of an equation.

Solve multi-step linear equations in one variable.

Solve linear equations with variables on both sides and distinguish equations having one, no, or infinitely many solutions.

Rearrange a literal equation to isolate a specified variable.

Solve linear absolute-value equations by separating the relevant cases.

Solve a two-variable linear system by substitution or elimination and classify its solution set.

Solve linear inequalities and express their solution sets using inequality notation, interval notation, and number-line graphs, accounting for reversal under multiplication or division by a negative number.

Solve compound linear inequalities involving conjunctions or disjunctions.

Identify mathematical propositions and determine their truth values in a given context.

Construct truth tables for negation, conjunction, disjunction, implication, and biconditional statements.

Translate mathematical statements between natural language and symbolic form using universal and existential quantifiers.

Negate statements containing universal or existential quantifiers.

Distinguish a conditional statement from its converse, inverse, and contrapositive.

Disprove a universal claim by constructing a counterexample.

Write a direct proof from the definitions of even and odd integers.

Write a proof by contraposition for an elementary statement about integer parity.

Simplify expressions using the product, quotient, and power laws for positive integer exponents.

Prove a selected positive-integer exponent law from the definition of exponentiation.

Simplify expressions containing zero and negative integer exponents while stating the required nonzero restrictions.

Convert between rational exponents and radical notation over the real numbers, accounting for domain restrictions.

Simplify radical expressions by extracting perfect powers under stated domain assumptions.

Perform arithmetic with simplified radical expressions.

Prove that the square root of 2 is irrational by contradiction.

Solve radical equations and reject extraneous candidate solutions.

Determine whether a table, mapping, graph, or equation defines a function.

Determine the domain and range of a function from a formula or graph.

Evaluate functions, including piecewise-defined functions.

Graph a linear function and determine its slope and intercepts.

Construct the equation of a linear function from suitable point or slope information.

Graph a transformed function using translations, reflections, and scalings.

Compute the composition of two functions and determine its domain.

Determine whether a simple function is one-to-one and find and verify its inverse by composition when an inverse exists.

Write a polynomial in standard form and identify its degree and leading coefficient.

Add and subtract polynomials.

Multiply polynomials using the distributive property.

Prove a special-product identity by applying the distributive property.

Factor polynomials by extracting a greatest common factor or grouping terms.

Factor monic quadratic trinomials over the integers.

Factor nonmonic quadratic trinomials over the integers.

Factor polynomials using difference-of-squares, perfect-square, and sum-or-difference-of-cubes identities.

Solve factorable polynomial equations using the zero-product property.

Rewrite a quadratic expression in vertex form by completing the square.

Derive the quadratic formula by completing the square in a general quadratic equation.

Solve quadratic equations using the quadratic formula.

Use the discriminant to classify the roots of a real quadratic equation.

Represent complex numbers in rectangular form and perform arithmetic with them.

Solve quadratic equations whose solutions are nonreal complex numbers.

Determine the vertex, axis, intercepts, and graph of a quadratic function.

Divide polynomials using long division.

Divide a polynomial by a linear divisor using synthetic division.

Apply the remainder theorem to evaluate a polynomial.

Apply the factor theorem to test whether a linear expression is a factor.

Use the rational root theorem to list possible rational zeros of an integer polynomial.

Use the fundamental theorem of algebra to count complex roots with multiplicity.

Factor and solve higher-degree polynomial equations by combining root and division theorems.

Relate the multiplicity of a real zero to the local behavior of a polynomial graph.

Determine the domain restrictions of a rational expression.

Simplify rational expressions while retaining their original domain restrictions.

Multiply and divide rational expressions.

Add and subtract rational expressions using a least common denominator.

Simplify complex rational expressions.

Prove a rational identity on its stated domain.

Solve rational equations and check candidates against domain restrictions.

Solve rational inequalities using sign analysis.

Graph exponential functions and distinguish exponential growth from decay.

Solve exponential equations by rewriting both sides with a common base.

Define logarithms as inverses of exponential functions and convert between logarithmic and exponential forms.

Derive the product, quotient, and power laws of logarithms from exponent laws.

Expand and condense logarithmic expressions while respecting domain restrictions.

Solve exponential equations using logarithms.

Solve logarithmic equations while enforcing domain restrictions.

Construct and solve an exponential growth or decay model from given data.

Represent sets using roster and set-builder notation and interpret membership and subset symbols.

Compute unions, intersections, differences, and complements of sets.

Prove a subset relation or set equality by reasoning about arbitrary elements.

Form Cartesian products and represent binary relations as subsets of Cartesian products.

Test a relation for reflexivity, symmetry, and transitivity and determine whether it is an equivalence relation.

Determine whether a function is injective, surjective, or bijective.

Construct the inverse of a bijection and prove that it is an inverse.

Prove an elementary algebraic or divisibility statement by mathematical induction.

Apply the definition of divisibility to prove elementary statements about integers.

Use the division algorithm to express an integer in quotient-remainder form.

Compute a greatest common divisor using the Euclidean algorithm.

Express a greatest common divisor as an integer linear combination using the extended Euclidean algorithm.

Prove that congruence modulo an integer is an equivalence relation.

Prove that addition and multiplication on residue classes modulo an integer are well defined.

Solve a linear congruence using greatest common divisors and modular inverses.

Verify that the residue classes modulo an integer form an abelian group under addition by checking the group axioms.