DSST Fundamentals of College Algebra flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Imaginary unit i

The number i is defined by i² = −1. For any positive real a, √(−a) = i√a. Powers of i cycle every four: i, −1, −i, 1.

Complex number form

A complex number is written a + bi, where a is the real part and b is the imaginary part (both real numbers). Equality holds iff real parts and imaginary parts match.

Complex conjugate

The conjugate of a + bi is a − bi. Their product is a² + b², a real number. Conjugates are used to divide complex numbers and to rationalize denominators.

Modulus of complex number

For z = a + bi, the modulus is |z| = √(a² + b²). It equals the distance from the origin to the point (a, b) in the complex plane.

Multiplying complex numbers

Use FOIL: (a+bi)(c+di) = ac + adi + bci + bdi². Replace i² with −1 to get (ac − bd) + (ad + bc)i.

Dividing complex numbers

Multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) = [(a+bi)(c−di)]/(c²+d²). The result is in standard form.

Powers of i

For integer n > 0, iⁿ depends on n mod 4: remainder 1 gives i, 2 gives −1, 3 gives −i, 0 gives 1. Example: i²⁵ = i¹ = i.

Adding complex numbers

Add real parts and imaginary parts separately: (a+bi) + (c+di) = (a+c) + (b+d)i. Subtraction works the same way with signs.

Order of operations

Evaluate expressions in this order: grouping symbols, exponents, multiplication and division left to right, then addition and subtraction left to right.

Exponent rules

For a ≠ 0: aᵐ·aⁿ = aᵐ⁺ⁿ; aᵐ/aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ.

Quadratic formula

The solutions of ax² + bx + c = 0 (a ≠ 0) are x = [−b ± √(b² − 4ac)]/(2a). This formula works for all quadratic equations.

Discriminant

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. If D > 0, two distinct real roots; D = 0, one real double root; D < 0, two complex conjugate roots.

Difference of squares

a² − b² factors as (a − b)(a + b). This pattern is used to simplify expressions and solve equations quickly.

Sum and difference of cubes

a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²). These factorizations are useful in polynomial equations.

Remainder theorem

When a polynomial P(x) is divided by x − r, the remainder is P(r). If P(r) = 0, then x − r is a factor of P(x).

Logarithm rules

For positive M, N and base b > 0, b ≠ 1: log_b(MN) = log_b M + log_b N; log_b(M/N) = log_b M − log_b N; log_b(Mᵖ) = p log_b M.

Rationalizing denominators

To eliminate a radical from a denominator, multiply numerator and denominator by the radical or by the conjugate of the denominator, then simplify.

Linear equation solution

An equation ax + b = 0 with a ≠ 0 has exactly one solution x = −b/a. If a = 0 and b ≠ 0, there is no solution; if a = 0 and b = 0, every number is a solution.

Absolute value equation

For c ≥ 0, |u| = c is equivalent to u = c or u = −c. If c < 0, the equation has no solution because absolute value is never negative.

Absolute value inequality

For c > 0: |u| < c means −c < u < c; |u| > c means u < −c or u > c. These translate to compound inequalities without absolute value.

Inequality sign reversal

Adding or subtracting the same number on both sides of an inequality preserves its direction. Multiplying or dividing both sides by a negative number reverses the inequality sign.

Extraneous solutions

When solving rational or radical equations, operations like squaring or multiplying by an expression can introduce solutions that do not satisfy the original equation. Always check answers in the original.

Sum and product of roots

For ax² + bx + c = 0 with roots r and s, r + s = −b/a and r·s = c/a. These relationships help check solutions and form equations from roots.

Solving exponential equations

If possible, express both sides with the same base and equate exponents. Otherwise, take the logarithm of both sides and solve for the variable.

Interval notation

A solution set can be written with intervals: (a, b) excludes endpoints; [a, b] includes them; (a, ∞) means x > a; (−∞, b] means x ≤ b.

Function definition

A function f from a set X to a set Y assigns to each element x in X exactly one element y in Y. The vertical line test identifies functions from their graphs.

Domain of a function

The domain is the set of all input values x for which the function is defined. Exclude values that make a denominator zero or an even root negative.

Even and odd functions

f is even if f(−x) = f(x) for all x in its domain; its graph is symmetric about the y-axis. f is odd if f(−x) = −f(x); its graph is symmetric about the origin.

Function composition

The composition (f ∘ g)(x) = f(g(x)) is defined when x is in the domain of g and g(x) is in the domain of f. Composition is not generally commutative.

Inverse function

If f is one-to-one, its inverse f⁻¹ satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. The graph of f⁻¹ is the reflection of the graph of f across the line y = x.

Horizontal line test

A function is one-to-one (and therefore has an inverse function) if and only if no horizontal line intersects its graph more than once.

Vertex of a parabola

For f(x) = ax² + bx + c (a ≠ 0), the vertex occurs at x = −b/(2a). The parabola opens upward if a > 0 and downward if a < 0.

Graph transformations

For y = a f(x − h) + k: h shifts horizontally (right if h > 0), k shifts vertically, a stretches or compresses vertically, and negative a reflects across the x-axis.