Algebra 1: Properties and Operations — Vocabulary Flashcards

Vocabulary flashcards covering key concepts from the lecture notes.

Commutative Property

For real numbers, changing the order of addends or factors does not change the result: a + b = b + a and a × b = b × a.

Associative Property

For real numbers, changing the grouping of terms does not affect the result: (a + b) + c = a + (b + c) and (ab)c = a(bc).

Identity Property (Addition)

Adding zero leaves a number unchanged: a + 0 = a.

Identity Property (Multiplication)

Multiplying by one leaves a number unchanged: a × 1 = a.

Distributive Property

Multiplication distributes over addition: a(b + c) = ab + ac.

Natural Numbers

Counting numbers used for counting: 1, 2, 3, … (positive integers).

Integers

Natural numbers, their negatives, and zero: …, -2, -1, 0, 1, 2, ….

Rational Numbers

Numbers that can be written as a ratio of integers (fractions), with a nonzero denominator.

Irrational Numbers

Real numbers that cannot be written as a ratio of two integers; decimal expansions are nonrepeating and nonterminating.

Fraction Multiplication Rule

Multiply numerators and denominators separately: (a/b)·(c/d) = (ac)/(bd).

Fraction Division Rule

Divide by a fraction by multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b)·(d/c) = (ad)/(bc).

Fractions with Same Denominator

Add or subtract fractions with the same denominator by combining numerators: (a/b) ± (c/b) = (a ± c)/b.

Fractions with Different Denominators

To add, find a common denominator, convert, then add the numerators.

Absolute Value

Distance from zero; always nonnegative: |a| ≥ 0; |a| = |-a|.

Absolute Value Properties

Multiplicative: |ab| = |a||b|; Quotient: |a/b| = |a|/|b|.

Exponent

A number of times a base is used as a factor: a^n means a multiplied by itself n times (n ≥ 0; extendable to negative).

Product Rule for Exponents

When bases are the same: a^m · a^n = a^(m+n).

Power Rule for Exponents

Raising a power to another power: (a^m)^n = a^(mn).

Quotient Rule for Exponents

Dividing powers with the same base: a^m / a^n = a^(m−n) (a ≠ 0).

Negative Exponent Rule

A negative exponent indicates a reciprocal: a^(-n) = 1/(a^n).

Product to a Power

Raising a product to a power: (ab)^n = a^n b^n.

Quotient to a Power

Raising a quotient to a power: (a/b)^n = a^n / b^n.

Rational Exponents

Exponents that are fractions: a^(m/n) = (n-th root of a)^m, with conditions on a when n is even.

nth Root

The radical form of a^(1/n); the n-th root of a.

Radical Exponent Form

Expressing roots as fractional exponents: a^(1/n) = √[n]{a}.

Rationalizing the Denominator

Eliminating a radical in the denominator by multiplying numerator and denominator by a suitable value.

GCF (Greatest Common Factor)

The largest factor common to all terms of a polynomial.

Factoring by GCF

Factor out the greatest common factor from each term.

Difference of Squares

A^2 − B^2 = (A − B)(A + B).

Sum of Cubes

A^3 + B^3 = (A + B)(A^2 − AB + B^2).

Difference of Cubes

A^3 − B^3 = (A − B)(A^2 + AB + B^2).

FOIL

First, Outer, Inner, Last—method for multiplying two binomials.

Monomial, Binomial, Trinomial

Monomial: one term; Binomial: two terms; Trinomial: three terms.

Prime Polynomial

A polynomial that cannot be factored further over the integers.

Factoring by Grouping

Group terms to factor common factors and then factor the resulting expression.

Quadratic Factoring (ax^2 + bx + c)

Factorization technique for quadratics by finding two numbers that multiply to ac and sum to b.

Equations

A statement that two expressions are equal; solving finds the value(s) of the variable that satisfy it.

Root/Solution

The value(s) of the unknown that satisfy the equation.

Linear Equation

An equation of the form ax + b = 0; its solution is x = −b/a (when a ≠ 0).

Domain

Set of all input values (x-values) for which a function is defined.

Range

Set of all output values (y-values) a function can take.

Function

A relation that assigns exactly one output for every input; denoted y = f(x) and described by its domain and range.

Function Evaluation

Computing the value f(a) by substituting a into the function.

Translating Verbal Expressions

Converting words into algebraic expressions (e.g., twice a number plus five becomes 2x + 5).

Distance, Rate, Time (D = RT)

Relationship to compute distance, speed, or time: Distance = Rate × Time.

Mixture Problems

Combine solutions with different concentrations: C1V1 + C2V2 = CfVt (final concentration times total volume).

Simple Interest

I = Prt; P = principal, r = rate (as a decimal), t = time (in years).

Work Problems (Time)**

If two tasks are done simultaneously: 1/tA + 1/tB = 1/tT, where tT is combined time.