Pre calc lec 4

Introductory Example

• Concept of Removal: To clarify combining like terms, consider removing objects:

  • Example: Removing 2 scissors, 3 dice, and 5 green pens from collections.

Like Terms

• Definition: Like terms must share the same variable and exponent.

• Example with Variables:

  • Scissors: x^4

  • Dice: x^2

  • Pens: x

  • Calculation: Combining 3 scissors (3x^4), 6 dice (6x^2), and 7 pens (7x) with removals (2x^4, 3x^2, 5x).

Combining Like Terms

• Process:

  • Combine terms with the same variable and exponent.

  • Example expression: x * (x + 2) - 7x * (x - 4)

• Distributing:

  • Distribute x:

    • x * x = x^2

    • x * 2 = 2x

  • Distribute -7x:

    • -7x * x = -7x^2

• Combining Results:

  • Collect x^2 terms to simplify: 1x^2 - 7x^2 = -6x^2.

Binomial Multiplication

• Expression: (x + 2)(3x - 1)

• Action: All parts of the first binomial interact with the second:

  • First: x * 3x = 3x^2

  • Middle: 2 * 3x = 6x

  • Last: 2 * (-1) = -2

• Result: Combined gives 3x^2 + 6x - 2.

Squaring a Binomial

• Concept: Squaring involves multiplying the binomial by itself.

• FOIL Method:

  • Multiply First: x * x

  • Outer: x * 2

  • Inner: 2 * x

  • Last: 2 * 2

Factoring Concepts

• Definition: Opposite of distributing; reverts a polynomial.

• Example: Factor 18x^3 + 27x^2

  • Identify greatest common factor (GCF):

    • GCF of 18 & 27 = 9

    • For x terms: X^2 can be pulled out (smallest degree).

  • Result: 9x^2(2x + 3).

Verifying Factorization

• Process: Check correctness by distributing back.

• Example:

  • 9x^2(2x + 3) → 18x^3 + 27x^2 (confirms factorization).

Factoring Quadratic Expressions

• Consider x^2 + 6x + 8:

  • Goal: Transform into binomials, needing two numbers that add to 6 and multiply to 8.

  • Find: (x + 2)(x + 4).

• Y-intercept: Plugging in x = 0 gives y = 8 (easily computed).

• Finding Roots: Set factors to 0:

  • Solve (x + 2) = 0 leads to x = -2

  • Solve (x + 4) = 0 leads to x = -4

Solving Quadratic Equations via Factoring

• Given: x^2 + 10x - 24 = 0

  • Move constant to one side by subtracting 24:

  • Factor to obtain two expressions equal to 0.

• Example Roots: x = 2 and x = -12 after solving.

Solving Inequalities

• Similar to equality except:

  • Operations on inequalities maintain direction unless multiplied by a negative.

• Example: For -3x ≤ -18, divide by -3 (flip inequality): x ≥ 6.

Summary of Key Concepts

• Identifying like terms is crucial for simplifying polynomials.

• Processes for multiplying, distributing, factoring, and solving equations are fundamental in algebra.