Factoring Trinomials Basics

Introduction

• Objective: Learn to factor trinomials in the form of ( ax^2 + bx + c ) where ( a = 1 ).
• Context: Transition from expanded form to factored form after learning to multiply binomials.

Trinomial Form with Leading Coefficient 1

• Form: ( x^2 + bx + c )
• Process:
  1. Set up two parentheses.
  2. Place factors of ( x^2 ) (which are ( x ) and ( x )) in the first position of each parenthesis.
  3. Find factors of ( c ) that add up to ( b ).

Example 1: ( x^2 + 8x + 12 )

• Steps:
  • Create parentheses: (x )(x ).
  • Find factors of 12 that add to 8: ( 2 \times 6 ).
  • Resulting factors: ( (x + 2)(x + 6) ) or ( (x + 6)(x + 2) ).

Example 2: ( x^2 - 4x - 32 )

• Steps:
  • Create parentheses: (x )(x ).
  • Factors of -32 that add to -4: ( -8 \times 4 ).
  • Resulting factors: ( (x - 8)(x + 4) ).

Example 3: ( y^2 - 13y + 36 )

• Steps:
  • Create parentheses: (y )(y ).
  • Factors of 36 that add to -13: ( -4 \times -9 ).
  • Resulting factors: ( (y - 4)(y - 9) ).

Example 4: ( x^2 + x - 12 )

• Steps:
  • Create parentheses: (x )(x ).
  • Factors of -12 that add to 1: ( -3 \times 4 ).
  • Resulting factors: ( (x - 3)(x + 4) ).

Example 5: ( x^2 + 28x + 75 )

• Steps:
  • Create parentheses: (x )(x ).
  • Factors of 75 that add to 28: ( 3 \times 25 ).
  • Resulting factors: ( (x + 3)(x + 25) ).

Factoring with the Greatest Common Factor (GCF)

Example 6: ( 3x^2 + 27x + 42 )

• Steps:
  • Factor out 3: ( 3(x^2 + 9x + 14) ).
  • Factors of 14 that add to -9: ( -2 \times -7 ).
  • Resulting factors: ( 3(x - 2)(x - 7) ).

Example 7: ( 2y^3 - 12y^2 - 80y )

• Steps:
  • Factor out 2y: ( 2y(y^2 - 6y - 40) ).
  • Factors of -40 that add to -6: ( -10 \times 4 ).
  • Resulting factors: ( 2y(y - 10)(y + 4) ).

Conclusion

• Summary: The process involves identifying appropriate factors and paying attention to signs.
• Tips: Utilize multiplication tables or lists to aid in identifying factor pairs.
• Note: Always check the greatest common factor before proceeding with factoring.