Factoring Trinomials the Easy Way

Introduction

• Focus on factoring trinomials the easy way.
• Examples of factoring expressions where the leading coefficient is 1 and when it's not 1.

Factoring Trinomials with Leading Coefficient 1

Example 1: (x^2 + 5x + 6)

• Objective: Find two numbers that multiply to 6 but add to 5.
• Solution:
  • Numbers that multiply to 6: 1 & 6, 2 & 3.
  • 2 + 3 = 5.
  • Factored form: ((x + 2)(x + 3)).

Example 2: (x^2 + 8x + 15)

• Objective: Find two numbers that multiply to 15 but add to 8.
• Solution:
  • Numbers that multiply to 15: 1 & 15, 3 & 5.
  • 3 + 5 = 8.
  • Factored form: ((x + 3)(x + 5)).

Example 3: (x^2 - 7x + 12)

• Objective: Find two numbers that multiply to 12 but add to -7.
• Solution:
  • Numbers that multiply to 12: 1 & 12, 2 & 6, 3 & 4.
  • -3 + (-4) = -7.
  • Factored form: ((x - 3)(x - 4)).

Example 4: (x^2 + 3x - 40)

• Objective: Find two numbers that multiply to -40 but add to 3.
• Solution:
  • Numbers that multiply to -40: -5 & 8.
  • -5 + 8 = 3.
  • Factored form: ((x - 5)(x + 8)).

Factoring Trinomials with Leading Coefficient Not 1

Example 5: (2x^2 - 3x - 2)

• Objective: Factor and solve for x if equal to 0.
• Solution:
  1. Multiply first and last term: 2 * (-2) = -4.
  2. Replace middle term: -3x with -4x + 1x.
  3. Factor by grouping:
     • 2x(x - 2) + 1(x - 2).
  4. Factored form: ((x - 2)(2x + 1)).
  5. Solve: (x = 2), (x = -\frac{1}{2}).

Example 6: (3x^2 + 8x - 3)

• Objective: Factor and solve for x.
• Solution:
  1. Multiply first and last terms: 3 * (-3) = -9.
  2. Numbers: -1 and 9 add to 8.
  3. Factor by grouping:
     • 3x(x + 3) - 1(x + 3).
  4. Factored form: ((x + 3)(3x - 1)).
  5. Solve: (x = -3), (x = \frac{1}{3}).

Example 7: (4x^2 - 4x - 3)

• Objective: Factor and solve for x.
• Solution:
  1. Multiply first and last terms: 4 * (-3) = -12.
  2. Numbers: 2 and -6 add to -4.
  3. Factor by grouping:
     • 2x(2x + 1) - 3(2x + 1).
  4. Factored form: ((2x + 1)(2x - 3)).
  5. Solve: (x = -\frac{1}{2}), (x = \frac{3}{2}).

Using the Quadratic Formula

• Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
• Apply the formula to solve quadratic when factoring is complex.
• Example: (4x^2 - 4x - 3) can also be solved using the quadratic formula.

Conclusion

• Factoring trinomials involves finding two numbers that work with the constant and middle terms.
• When leading coefficient is not 1, use factoring by grouping or the quadratic formula for solutions.