Lecture on Factoring

Introduction to Factoring

• Factoring: The process of breaking down an expression into simpler terms that can be multiplied together to get the original expression.
• Greatest Common Factor (GCF): The largest factor that divides two numbers.

Finding the Greatest Common Factor

• Example: Expression 3x + 15

  • GCF is 3.
  • Factor: 3(x + 5)

• Example: Expression 7x - 28

  • GCF is 7.
  • Factor: 7(x - 4)

• Example: Expression 4x^2 + 8x

  • GCF is 4x.
  • Factor: 4x(x + 2)

• Example: Expression 5x^2 - 15x^3

  • GCF is 5x^2.
  • Factor: 5x^2(1 - 3x)

Factoring by Grouping

• Used for polynomials with four terms.

• Example: x^3 - 4x^2 + 3x - 12

  • Group: (x^3 - 4x^2) + (3x - 12)
  • GCF of first group: x^2
  • GCF of second group: 3
  • Factor: (x - 4)(x^2 + 3)

• Example: 2r^3 - 6r^2 + 5r - 15

  • Group: (2r^3 - 6r^2) + (5r - 15)
  • GCF of first group: 2r^2
  • GCF of second group: 5
  • Factor: (r - 3)(2r^2 + 5)

Factoring Trinomials

Leading Coefficient is 1

• Method: Find two numbers that multiply to the constant term and add to the middle term.

• Example: x^2 + 7x + 12

  • Numbers: 3 and 4.
  • Factor: (x + 3)(x + 4)

• Example: x^2 + 3x - 28

  • Numbers: 7 and -4.
  • Factor: (x + 7)(x - 4)

Leading Coefficient is not 1

• Method: Multiply leading coefficient by constant, then find numbers as above.

• Example: 2x^2 + 20x + 48

  • GCF: 2
  • Factor out: 2(x^2 + 10x + 24)
  • Factor: 2(x + 4)(x + 6)

Perfect Square Trinomials

• Form: a^2 + 2ab + b^2 = (a + b)^2

• Example: x^2 + 8x + 16

  • Factor: (x + 4)^2

• Example: 4x^2 + 12x + 9

  • Factor: (2x + 3)^2

Difference of Squares

• Form: a^2 - b^2 = (a + b)(a - b)

• Example: x^2 - 25

  • Factor: (x + 5)(x - 5)

Factoring Sums and Differences of Cubes

• Sum of Cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)

• Difference of Cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

• Example: x^3 + 8

  • Factor: (x + 2)(x^2 - 2x + 4)

Solving Equations by Factoring

• Example: 6x^2 - 30x = 0

  • GCF: 6x
  • Factor: 6x(x - 5) = 0
  • Solutions: x = 0, x = 5

• Example: 3x^2 - 27 = 0

  • GCF: 3
  • Factor: 3(x^2 - 9) = 0
  • Solutions: x = 3, x = -3

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• Practice problems include using these techniques to factor expressions like trinomials, perfect squares, and cubes.
• Always check work by multiplying back out to verify the factored form is correct.