Factoring Polynomials Overview

Factoring Order

• Factoring follows a specific order: GCF → DOTS → TRI → AC → QF.

Greatest Common Factor (GCF)

• Formula: ab + ac = a(b + c).

Difference of Two Perfect Squares (DOTS)

• Formula: x² - y² = (x + y)(x - y).

Trinomial Factoring (TRI)

• Example: x² - x + 6 = (x + 2)(x - 3).

"AC" Method / Earmuff Method

• Used when a ≠ 1.
• Example: For 2x² + 15x + 18, factors are found by transforming to x² + 15x + 36 → (x + 12)(x + 3).

Quadratic Formula

• Used to find the roots of quadratics when other methods fail: x = -b ± √(b² - 4ac) / 2a.

Dividing Polynomials

• Division Algorithm: Dividend/Divisor = Quotient + Remainder/Divisor.

Long Division of Polynomials

• This method involves dividing the polynomial as you would with numbers, aligning terms.

Synthetic Division of Polynomials

• A shorthand method for dividing by linear factors.

Factor by Grouping

• Method to simplify polynomials by grouping terms and factoring out common factors.

Factoring Perfect Cubes

• Use SOAP: Same sign, Opposite sign, Always Positive.

Remainder Theorem

• When f(x) is divided by (x-a), the remainder equals f(a).

Factor Theorem

• If f(a) = 0, then (x - a) is a factor of f(x).

Quadratic Definition

• A polynomial equation of degree two.

Standard Form of a Quadratic Equation

• Structure: ax² + bx + c = 0 with a, b, c constants and a ≠ 0.

Roots of a Quadratic

• Sum: r₁ + r₂ = -b/a.
• Product: r₁ ∙ r₂ = c/a.

Graph of Quadratic

• Key points: X-intercepts, Turning Point (Vertex), Axis of Symmetry (x = c), Focus, Directrix.

The Discriminant

• Determines the nature of roots: b² - 4ac.

Definition of a Function

• A relation where each x-value connects to exactly one y-value.

Domain

• The set of all possible x-values, subject to restrictions like non-zero denominators or non-negative radicands.

Range

• The set of all possible y-values of a function.

Composition of Functions

• Combining functions, written as f(g(x)), calculated right to left.

One-to-One Function

• A function with no repeating x or y values, passing both horizontal and vertical line tests.

Inverse Functions

• Reflect the original function over y = x; only one-to-one functions have inverses.

End Behavior

• Determined by the degree and leading coefficient of the polynomial.

Multiplicity

• The number of times a polynomial's root occurs.

Odd Degree Polynomials Behavior

• Positive Coefficient: f(x) → ∞ as x → ∞; f(x) → -∞ as x → -∞.
• Negative Coefficient: f(x) → -∞ as x → ∞; f(x) → ∞ as x → -∞.

Even Degree Polynomials Behavior

• Positive Coefficient: f(x) → ∞ on both ends.
• Negative Coefficient: f(x) → -∞ on both ends.

Complex Numbers

• Imaginary unit i satisfies i² = -1.

Rational Expressions and Equations

• Addition/Subtraction requires a common denominator; multiplication involves factoring, reducing, and multiplying.

Properties of Exponents

• x⁰ = 1.

Converting Radians to Degrees

• Multiply radians by π/180.

Converting Degrees to Radians

• Multiply degrees by 180/π.

Trigonometric Ratios

• sin θ = opposite/hypotenuse.
• cos θ = adjacent/hypotenuse.
• tan θ = opposite/adjacent.
• csc θ = hypotenuse/opposite.
• sec θ = hypotenuse/adjacent.
• cot θ = adjacent/opposite.

Reciprocal Functions

• Examples include: sin θ = 1/csc θ, sin 0 = 1/sin 0, cos θ = 1/sec θ.

Arc Length of a Circle

• Formula: s = r∙θ, where s is the arc length, r is the radius, and θ is in radians.

Unit Circle

• A circle with a radius of 1 centered at the origin.

Special Right Triangles

• Include angle-based (45°-45°-90°) and side-based (3:4:5) configurations for simpler calculations.

Trigonometric Graphs

• Graphs representing sine, cosine, and tangent functions with specific characteristics.

Pythagorean Identities

• Fundamental trigonometric identities: sin²θ + cos²θ = 1, tan²θ + 1 = sec²θ, 1 + cot²θ = csc²θ.

Inverse Trig Functions

• Notation: sin⁻¹x for the inverse of sin, cos⁻¹x for the inverse of cos, tan⁻¹x for the inverse of tan.

Sigma Notation

• Represents the sum of terms with a common form.

Finite Sequences

• Arithmetic and Geometric series have specific formulas to calculate their sums.

Statistics: Surveys and Experiments

• Surveys gather opinions; observational studies do not interact with subjects; controlled experiments analyze effects of conditions.

Probability Concepts

• Independent events do not affect each other; dependent events do. z-scores measure distances from mean; complementary events are calculated using P(A') = 1 - P(A).

Mutually Exclusive Events

• For mutually exclusive A and B, P(A or B) = P(A) + P(B). If not mutually exclusive, P(A or B) = P(A) + P(B) - P(A and B).