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)?

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

What is the formula for the 'AC' Method?

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

What is the Quadratic Formula?

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

What does the Division Algorithm state?

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

What is the Remainder Theorem?

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

What is the Factor Theorem?

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

What is a quadratic equation?

<p>A polynomial equation with a degree of two.</p> Signup and view all the answers

What is the standard form of a quadratic equation?

<p>ax² + bx + c = 0</p> Signup and view all the answers

What is the sum of the roots of a quadratic?

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

What is the product of the roots of a quadratic?

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

What is the discriminant?

<p>b² - 4ac</p> Signup and view all the answers

What is a function?

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

What is the domain of a function?

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

What is the range of a function?

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

What are one-to-one functions?

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

What is end behavior in graphs?

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

What is multiplicity in polynomials?

<p>How many times a particular number is a zero.</p> Signup and view all the answers

What is the formula for arc length of a circle?

<p>s = r∙θ</p> Signup and view all the answers

What are the Pythagorean identities?

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

What is a survey in statistics?

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

What is an observational study?

<p>The observer examines results without interaction.</p> Signup and view all the answers

What is a controlled experiment?

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

What is conditional probability?

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

What does it mean for events to be mutually exclusive?

<p>If A and B cannot occur at the same time.</p> Signup and view all the answers

What is the formula for the confidence interval?

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

What is a z-score?

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

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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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