Factorising Non-monic Trinomials

Factorising Non-monic Trinomials

Method for Factorising ax^2 + bx + c

• Objective: Factorise a quadratic expression of the form ax^2 + bx + c.

• Steps:

  1. Find ac: Multiply the coefficient of x^2 (which is a) by the constant term c.

  2. Find factors of ac: Identify two factors of ac that add up to b (the coefficient of x).

  3. Split b: Rewrite the middle term bx as the sum of two terms using the factors found in the previous step.

  4. Factorise by grouping: Group the terms in pairs and factorise each pair. Then, factor out the common binomial factor.

Example 1: Factorise 6x^2 + 17x + 5

• a = 6, b = 17, c = 5

• Find ac: ac = 6 \times 5 = 30

• Find factors of 30 that add to 17: 15 \times 2 = 30, 15 + 2 = 17

• Split b: 6x^2 + 17x + 5 = 6x^2 + 2x + 15x + 5

• Factorise by grouping:

  • 2x(3x + 1) + 5(3x + 1)

  • (3x + 1)(2x + 5)

Example 2: Factorise 4x^2 - 4x - 15

• a = 4, b = -4, c = -15

• Find ac: ac = 4 \times -15 = -60

• Find factors of -60 that add to -4: -10 + 6 = -4

• Split b: 4x^2 - 4x - 15 = 4x^2 - 10x + 6x - 15

• Factorise by grouping:

  • 2x(2x - 5) + 3(2x - 5)

  • (2x - 5)(2x + 3)

Note

• Factorising by inspection/trial and error is possible, but the method described above is helpful when dealing with large numbers where trial and error becomes difficult.

Practice Factorisation

1. Factorise the following:

   • a) 3x^2 + 8x + 4

   • b) 5x^2 + 12x + 4

   • c) 3x^2 + 29x + 18

   • d) 19x + 10x^2 + 6

   • e) 6x^2 - 19x + 10

   • f) 2x + 3x^2 - 8

   • g) 7x^2 + 26x - 8

   • h) 10x^2 - x - 3

   • g) 5x^2 - 32x + 12

   • h) 15x^2 - 26x + 8

Factoring with Common Factors

Initial Check

• Always begin by checking for common factors before attempting to factorise the trinomial.

Example

• Factorise 10x^2 - 16x + 6

  • Common factor: 2(5x^2 - 8x + 3)

  • Now, factorise 5x^2 - 8x + 3 (i.e., a = 5, b = -8, c = 3)

    • ac = 5 \times 3 = 15

    • 15 = -5 \times -3, -5 + -3 = -8

    • 2(5x^2 - 5x - 3x + 3) = 2(5x(x - 1) - 3(x - 1))

    • = 2(x - 1)(5x - 3)

More Factorisation Practice

2. Factorise the following:

   • a) 6x^2 + 32x + 10

   • b) 40x^2 + 110x + 60

   • c) 12x^2 + 45x + 27

   • d) 16x^2 + 16x - 12

   • e) 12x^2 + 58x + 70

   • f) -45x^2 + 33x + 36