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Factoring Trinomials with Leading Coefficients

Mar 9, 2025

Factoring Trinomials with a Leading Coefficient Not Equal to One

This lecture provides a method for factoring trinomials where the leading coefficient is not one. This method might not be often taught in school, particularly in the U.S., but it can be very useful.

General Steps for Factoring

  1. Multiply the Leading Coefficient by the Constant Term

    • Example 1: For the trinomial (12x^2 + 17x + 6), multiply 12 (leading coefficient) by 6 (constant term) to get 72.
    • The expression becomes (x^2 + 17x + 72).

  2. Factor the Transformed Quadratic

    • Find two numbers that multiply to the new constant (72) and add up to the middle coefficient (17).
    • Possible pairs: (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9).
    • The correct pair is (8, 9) as 8 + 9 = 17.
    • The factors are ((x + 8)) and ((x + 9)).

  3. Adjust the Factors

    • Divide each constant in the factors by the original leading coefficient, 12.
    • Simplify the fractions:
      • (\frac{8}{12} = \frac{2}{3}) and (\frac{9}{12} = \frac{3}{4}).
    • Rewrite the factors as ((x + \frac{2}{3})) and ((x + \frac{3}{4})).

  4. Clear the Denominators

    • Multiply the denominator by the corresponding x term:
      • (3(x + \frac{2}{3}) = 3x + 2)
      • (4(x + \frac{3}{4}) = 4x + 3)
    • Final factors: ((3x + 2)(4x + 3)).

Second Example

  1. Starting Expression: (6x^2 - 5x - 4)

    • Multiply the leading coefficient 6 by the constant -4 to get -24.
    • Expression: (x^2 - 5x - 24).

  2. Factor the New Quadratic

    • Find two numbers that multiply to -24 and together add to -5.
    • Possible pairs (multiply to 24): (1, 24), (2, 12), (3, 8), (4, 6).
    • Correct pair: 3 and 8, since 8 - 3 = 5. Place larger number with negative sign as -8.
    • Rewrite as ((x - 8)) and ((x + 3)).

  3. Adjust and Simplify

    • Divide by original leading coefficient, 6:
      • (\frac{3}{6} = \frac{1}{2}) and (\frac{8}{6} = \frac{4}{3}).
    • Rewrite factors: ((x + \frac{1}{2})) and ((x - \frac{4}{3})).

  4. Clear Denominators

    • Multiply the denominator by the corresponding x term:
      • (2(x + \frac{1}{2}) = 2x + 1)
      • (3(x - \frac{4}{3}) = 3x - 4)
    • Final factors: ((2x + 1)(3x - 4)).

Conclusion

This method is effective for factoring trinomials with a leading coefficient that is not one. It involves transforming the trinomial for easy factoring, adjusting the factors by the original leading coefficient, and then clearing any fractions.

This approach can be especially helpful if you're preparing for problems in standardized testing contexts or need a straightforward method for complex trinomials.

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