DSST Fundamentals of College Algebra cheat sheet

Imaginary unit i satisfies i² = −1; for a > 0, √(−a) = i√a.

A complex number z = a + bi has real part a and imaginary part b; a is the real part, b the imaginary coefficient.

Addition/subtraction: (a+bi) ± (c+di) = (a±c) + (b±d)i.

Multiplication: (a+bi)(c+di) = (ac − bd) + (ad + bc)i; use FOIL and replace i² with −1.

Conjugate of a+bi is a−bi; product (a+bi)(a−bi) = a² + b² (a real number).

Division: multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) = [(a+bi)(c−di)]/(c²+d²).

Powers of i cycle with period 4: i¹=i, i²=−1, i³=−i, i⁴=1; for n>0, iⁿ = i^(n mod 4).

Equality: a+bi = c+di iff a=c and b=d.

Modulus |a+bi| = √(a²+b²); this is the distance from the origin to (a,b) in the complex plane.

Order of operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

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

Radicals: ⁿ√(aᵐ) = a^(m/n); ⁿ√(ab) = ⁿ√a · ⁿ√b; rationalize denominators by multiplying by the conjugate when needed.

Factoring patterns: a² − b² = (a−b)(a+b); a³ ± b³ = (a ± b)(a² ∓ ab + b²); perfect-square trinomials a² ± 2ab + b².

Quadratic formula: x = [−b ± √(b²−4ac)]/(2a) solves ax²+bx+c=0 (a≠0).

Discriminant D = b²−4ac: D>0 two distinct real roots; D=0 one real double root; D<0 two complex conjugate roots.

Polynomial division: dividend = divisor × quotient + remainder; Remainder Theorem: remainder of P(x) divided by (x−r) is P(r).

Factor Theorem: (x−r) is a factor of P(x) iff P(r)=0.

Logarithm rules: 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; log_b b = 1; log_b 1 = 0.

Linear equation ax+b=0 (a≠0) has unique solution x = −b/a.

Absolute value equation |u| = c (c≥0) is equivalent to u = c or u = −c; if c<0, no solution.

Absolute value inequality |u| < c (c>0) ⇔ −c < u < c; |u| > c ⇔ u < −c or u > c.

Quadratic equation ax²+bx+c=0: solve by factoring, completing the square, or quadratic formula; sum of roots = −b/a, product = c/a.

Rational equation: multiply both sides by the LCD, then check for extraneous solutions that make any denominator zero.

Radical equation: isolate the radical, square both sides, solve, and check for extraneous roots.

Linear inequality: adding/subtracting same quantity preserves direction; multiplying/dividing by a negative number reverses the inequality sign.

Compound inequality: solve each part separately and intersect the solution sets; express answers in interval notation.

Exponential equation: rewrite both sides with the same base, or take logarithms of both sides; logarithmic equation: convert to exponential form and check domain.

A function f assigns each x in its domain exactly one y; vertical line test: a graph is a function iff no vertical line meets it more than once.

Domain: set of allowable x-values; exclude values making a denominator zero or a radicand negative (for even roots).

Range: set of all possible y-values; find from graph or by analyzing the function.

Even function: f(−x)=f(x) for all x in domain; graph symmetric about the y-axis. Odd function: f(−x)=−f(x); graph symmetric about the origin.

Composition: (f∘g)(x)=f(g(x)); domain of f∘g is all x in domain of g with g(x) in domain of f.

Inverse function f⁻¹ reverses f: f(f⁻¹(x))=x and f⁻¹(f(x))=x; graph of f⁻¹ is reflection of graph of f across line y=x; f must be one-to-one.

Horizontal line test: a function is one-to-one (has an inverse function) iff no horizontal line intersects its graph more than once.

Transformations of y=f(x): y=f(x−h)+k shifts right h, up k; y=−f(x) reflects across x-axis; y=f(−x) reflects across y-axis; y=a f(x) stretches vertically by |a| (shrinks if 0<|a|<1).

Quadratic function f(x)=ax²+bx+c has vertex at x = −b/(2a); opens upward if a>0, downward if a<0; axis of symmetry x = −b/(2a).