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) $-45x^2 + 33x + 36$