Algebraic manipulation and vocabulary study guide
Algebra • Notation, vocabulary and manipulation
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Key concepts
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Algebraic expressions
An algebraic expression is a combination of numbers, variables and operations that represents a value. Terms are the parts of an expression separated by plus or minus signs; coefficients multiply variables and constants stand alone. Clear labelling of terms prevents incorrect combining of unlike terms and ensures correct simplification steps. Collecting like terms reduces the number of terms and reveals coefficients for further operations. Grouping like terms causes simplification, which makes substitution and comparison of expressions straightforward.
Equations and formulae
An equation states equality between two expressions and contains an unknown to solve for; a formula is a general equation that defines a relationship between variables. Rearrangement of a formula isolates a required variable, so substitution and evaluation become possible. The balance principle governs algebraic manipulation of equations: performing the same operation on both sides preserves equality. Solving linear equations produces numerical values for variables, while manipulating formulae produces expressions for a chosen variable. Clear notation of the subject variable prevents ambiguity during rearrangement.
Identities
An identity is an equality that holds for all allowed values of the variables concerned. Verification of an identity requires algebraic manipulation that transforms one side into the other, not substitution of particular values. Recognition of identities guides simplification choices and prevents false conclusions that arise from testing only special cases. Common identities, such as the difference of squares or square of a binomial, provide templates for expansion and factorisation. Application of an identity reduces manipulation steps and reveals structure in complex expressions.
Inequalities
An inequality compares two expressions and indicates a range of permissible values for variables. Manipulation of inequalities follows similar algebraic rules to equations, with a key limit: multiplication or division by a negative number reverses the inequality sign. Careful tracking of this reversal prevents sign errors and incorrect solution sets. Representation of solutions uses inequality notation or interval notation. Graphical number-line sketches convey cause → effect relations between algebraic manipulation and resulting solution ranges.
Terms, coefficients and factors
A term is a single product of numbers and variables; a coefficient multiplies the variable part of a term. A factor is a quantity multiplied with others to form a product; factorisation breaks an expression into factors. Correct identification of common factors causes factorisation to succeed and prevents illegal cancellation. Factorisation by common factor, grouping or formula application converts sums into products, so solving equations and simplifying fractions becomes possible. Misidentification of factors causes incorrect simplifications and invalid algebraic steps.
Simplifying expressions
Simplification removes redundancy and presents an expression in a standard, compact form. Collecting like terms, expanding brackets and cancelling common factors where permitted all contribute to simplification. Each simplification step reduces complexity, so subsequent algebraic procedures become easier and less error-prone. Limitations on simplification include non-like terms that cannot combine, and algebraic fractions that require factorisation before cancellation. Awareness of these limits prevents invalid operations.
Manipulation of surds
A surd is an irrational root left in root form to preserve exactness. Simplification of surds involves extracting integer factors from under the root and rationalising denominators when required. Rationalisation changes the form of an expression so that no surd appears in the denominator, which aids comparison and further algebraic work. Squaring and expanding expressions containing surds follow the same algebraic rules, with the added constraint that squaring introduces new rational terms. Careful tracking of signs and domain restrictions prevents introduction of extraneous solutions.
Algebraic fractions
An algebraic fraction contains polynomials in numerator and/or denominator. Simplification requires factorisation of numerator and denominator and cancellation of common factors, subject to excluding values that make the original denominator zero. Addition and subtraction of algebraic fractions require a common denominator, which causes expansion and combination of numerators. Division of algebraic fractions converts to multiplication by the reciprocal, so inversion of the divisor is the key step. Restrictions on variable values remain in force, and final statements of results note excluded values explicitly.
Key notes
Important points to keep in mind