Factoring Polynomials Overview

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Questions and Answers

What is the order of factoring?

DOTS (correct)

AC Method (correct)

Quadratic Formula (correct)

TRI (correct)

GCF (correct)

What is the formula for the Greatest Common Factor (GCF)?

ab + ac = a(b + c)

What is the formula for the Difference between Two Perfect Squares (DOTS)?

x² - y² = (x + y)(x - y)

What is the formula for a Trinomial (TRI)?

x² - x + 6 = (x + 2)(x - 3) Signup and view all the answers

What is the formula for the 'AC' Method?

AC (a≠1) 2x² + 15x + 18 Signup and view all the answers

What is the Quadratic Formula?

x = -b ± √b² - 4ac / 2a Signup and view all the answers

What does the Division Algorithm state?

Dividend/Divisor = Quotient + Remainder/Divisor Signup and view all the answers

What is the Remainder Theorem?

When f(x) is divided by (x - a), the remainder equals f(a). Signup and view all the answers

What is the Factor Theorem?

If f(a) = 0, then (x - a) is a factor of f(x). Signup and view all the answers

What is a quadratic equation?

A polynomial equation with a degree of two. Signup and view all the answers

What is the standard form of a quadratic equation?

ax² + bx + c = 0 Signup and view all the answers

What is the sum of the roots of a quadratic?

r₁ + r₂ = -b/a Signup and view all the answers

What is the product of the roots of a quadratic?

r₁ ∙ r₂ = c/a Signup and view all the answers

What is the discriminant?

b² - 4ac Signup and view all the answers

What is a function?

A relation where each x value is connected to a unique y value. Signup and view all the answers

What is the domain of a function?

The largest set of elements from the independent variable (x). Signup and view all the answers

What is the range of a function?

The set of elements for the dependent variable (y). Signup and view all the answers

What are one-to-one functions?

A function with no repeating x or y values. Signup and view all the answers

What is end behavior in graphs?

The direction a function heads at the ends of the graph. Signup and view all the answers

What is multiplicity in polynomials?

How many times a particular number is a zero. Signup and view all the answers

What is the formula for arc length of a circle?

s = r∙θ Signup and view all the answers

What are the Pythagorean identities?

sin²θ + cos²θ = 1; tan²θ + 1 = sec²θ; 1 + cot²θ = csc²θ. Signup and view all the answers

What is a survey in statistics?

Used to gather large quantities of facts or opinions. Signup and view all the answers

What is an observational study?

The observer examines results without interaction. Signup and view all the answers

What is a controlled experiment?

Two groups are studied with one receiving treatment and the other not. Signup and view all the answers

What is conditional probability?

The probability of an event given another event has occurred. Signup and view all the answers

What does it mean for events to be mutually exclusive?

If A and B cannot occur at the same time. Signup and view all the answers

What is the formula for the confidence interval?

A range of values used to estimate a population parameter. Signup and view all the answers

What is a z-score?

The number of standard deviations a value falls from the mean. Signup and view all the answers

Study Notes

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).

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Description

This quiz covers various methods of factoring polynomials, including the Greatest Common Factor (GCF), Difference of Two Perfect Squares (DOTS), Trinomial Factoring (TRI), and the Quadratic Formula. Understanding these concepts is crucial for mastering polynomial equations in algebra.