Lesson 6: Higher Factorisation

Objective

At the end of this lesson, students should be able to:

• Factorise and simplify more complex quadratic expressions.

Example 1

Factorise: y³ + 7y² + 12y

Solution

y³ + 7y² + 12y

Factorise y since it is common to all the terms.

y(y² +  7y + 12)

y² + 7y + 12  is a quadratic expression.

We require two numbers that will add to 7 and multiply to 12. The numbers are 3 and 4.

Y² + 7y + 12 = (y +3)(y + 4)

Therefore, y³ + 7y² + 12y = y (y + 3) (y + 4)

Example 2

Factorise; 24y² – 49x² – 5x²

Solution

24y² – 49x² – 5x²

– 49x² – 5x² = – 54x²

Therefore, 24y² – 49x² – 5x² = 24y² – 54x²

24y² – 54x²

6(4y² – 9x²) = 6 [ (2y)² – (3x)²]

= 6 (2y – 3x)(2y + 3x)

Example 3

Factorise: 3x (x – 3) + 4 (x – 7)

Solution

Expand first then simplify

3x (x – 3)  + 4 (x – 7)

3x² – 9x + 4x – 28

3x² – 5x – 28

We need two numbers that multiply to  -84 and add to – 5. They are -12 and 7

3x² – 12x + 7x – 28

3x (x – 4) + 7 (x – 4)

(x – 4) (3x + 7)

3x (x – 3) + 4 (x – 7) = (x – 4) (3x + 7)