Precalculus Review I (Video Notes)

Vocabulary flashcards covering key precalculus concepts from the lecture notes.

Real number line

A geometric representation of real numbers as points on a line with origin 0, where positive numbers lie to the right and negative numbers lie to the left; there is a one-to-one correspondence between real numbers and points, and the coordinate of a point is the associated real number.

Origin

The point on the real number line designated as 0 from which all coordinates are measured.

Coordinate (on the number line)

The real number associated with a point on the number line; the position relative to the origin.

Open interval

The set (a, b) of all real numbers x with a < x < b; endpoints a and b are not included.

Closed interval

The set [a, b] of all real numbers x with a ≤ x ≤ b; endpoints a and b are included.

Half-open interval

An interval that includes exactly one endpoint, such as (a, b] or [a, b).

Infinite interval (half-line)

Intervals that extend without bound, such as (a, ∞), [a, ∞), (−∞, a), (−∞, a].

Infinity notation

The symbol ∞, not a real number, used to denote unbounded limits in interval notation.

Exponent

The superscript n in b^n, indicating the base b is multiplied by itself n times.

Base

The number that is raised to a power in an exponent expression, e.g., in b^n, b is the base.

Power

The exponent n in b^n; indicates how many times the base is multiplied by itself.

Zero exponent

For b ≠ 0, b^0 = 1; note that 0^0 is undefined.

Nth root

The number b^(1/n) which, when raised to the nth power, gives b.

Monomial

An algebraic expression with one term of the form a x^m y^n where a is a real coefficient and m, n are nonnegative integers.

Polynomial

A sum of monomials; an expression formed by adding two or more monomials.

Degree of a polynomial

The highest total degree (sum of exponents) among the terms in the polynomial.

Like terms

Terms that have the same variable factors and can be combined by adding coefficients.

Distributive property

The rule ab + ac = a(b + c), used to remove parentheses and combine like terms.

Factoring

Expressing an algebraic expression as a product of factors; often begins by factoring out the greatest common factor.

Greatest Common Factor (GCF)

The largest factor common to all terms of an expression that can be factored out.

Difference of squares

A product formula: x^2 − y^2 = (x + y)(x − y).

Perfect-square trinomials

Trinomials of the form x^2 ± 2xy + y^2 = (x ± y)^2.

Sum of cubes

x^3 + y^3 = (x + y)(x^2 − xy + y^2).

Difference of cubes

x^3 − y^3 = (x − y)(x^2 + xy + y^2).

Quadratic equation

A polynomial equation of degree 2 in standard form ax^2 + bx + c = 0 with a ≠ 0.

Quadratic formula

Solutions x = (-b ± sqrt(b^2 − 4ac)) / (2a) for ax^2 + bx + c = 0 in standard form.

Roots of polynomial equations

The values of x that satisfy the polynomial equation; can be found by factoring or using the quadratic formula.

Rationalization

The process of eliminating radicals from the numerator or denominator of a fraction.