Algebra 2 Lecture Notes

Multiplying Binomials

Binomial Example: x + 3
Multiplying Binomials: Example: (x + 3) * (x - 2)
- Multiplies to create one number.
- Equation Example: (x + 3) * (x - 2) = 0
  - At least one factor must be zero.
  - Solutions: x = -3 or x = 2

FOIL: First, Outside, Inside, Last
Used for multiplying binomials:
- Example: (x + 3) * (x - 2)
- Calculation:
  - First: x * x = x^2
  - Outside: x * (-2) = -2x
  - Inside: 3 * x = 3x
  - Last: 3 * (-2) = -6
- Resulting in the trinomial: x^2 + x - 6
Reverse FOIL: Factor trinomials to binomials, e.g., x^2 + 8x + 12 = 0
- Match factors that add to middle term and multiply to constant term.
- Example solution: (x + 6) * (x + 2) = 0
- Solutions: x = -2, x = -6

Exponent Definition: How many times a number multiplies by itself.
- Example: 5^3 = 5 * 5 * 5 = 125
Basic Rules:
- Any number to the power of 1 is itself.
- a^0 = 1 (Multiplicative identity rule)
- Multiplication of powers: a^b * a^c = a^(b+c)*

Square Root: Number which, when multiplied by itself, yields the original number.
- Example: sqrt(16) = 4 and also -4
General Radicals:
- n-th root: a^(1/n)
- Example: 4th root of 16 = 2

Inverse Operations: Operations that undo one another.
- Example: f(x) = x + 4 has inverse f⁻¹(x) = x - 4
Complex Inverse Functions:
- Method: Replace variables and solve.
- Example: f(x) = 3x + 5
- Inverse function candidate if (f ∘ f⁻¹)(x) = (f⁻¹ ∘ f)(x) = x

Square Roots: Provide both positive and negative results.
Vertical Line Test: Demonstrates that certain equations are not true functions.
Even vs. Odd Powers:
- Even powers fail as inverse functions due to multiple outputs.
- Odd powers generally work properly for inverse functions.

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