Expansion and Factorisation

Expansion of Algebraic Expression

3x(y² + 8x)

When expanding the above expression, multiply each and every term inside the brackets by 3x.

= 3x (y²) + 3x (8x)

= 3xy² + 24x²

8yz(xy³ – 3x + 5)

= 8yz (xy³) – 8yz (3x) + 8yz (5)

= 8xy⁴z – 24xyz + 40yz

Make x as the subject of the formula

3y = 5 – x

3y + x = 5

x = 5 – 3y

2x + 2y = 5 + y – 3x

2x + 3x = 5 + y – 2y

5x = 5 – y

x = (5 – y)/5

y = 3/x

x = 3/y

2x² + 5y = 25

2x² = 25 – 5y

x² = (25 – 5y) /2

x = √ ( (25 –5y) / 2 )

Factorisation of Algebraic Expression

Factorising is “taking out common terms”, the reverse of expansion.

3xy² + 24x²

In constant terms (3 and 24), 3 is common.

In x terms (x and x²), x is common.

In y terms, nothing is in common.

3xy² + 24x² = 3x (y² + 8x)

Factorise:

• 12a + 3ab = 3a (4 + b)
• 12x²y – 4xy² = 4xy (3x – y)
• 2ab + 12abc + 6abcd = 2ab (1 + 6c + 3d)

Factorise: 2ac + 3ad – 6bc – 9bd

= 2ac + 3ad – 6bc – 9bd

= a (2c + 3d) – 3b (2c + 3d)

= (2c + 3d) (a – 3b)

Some Formulae

(a + b)² = a² + 2ab + b²

(a – b)² = a² – 2ab + b²

(a + b) (a – b) = a² – b²

These formulae may be used in Expansion and Factorisation.

Factorising Using Formula

x² + 4x + 4

= x² + 2 (x) (2) + 2²

= (x + 2)²

9x² + 24x + 16

= (3x)² + 2 (3x) (4) + 4²

= (3x + 4)²

x² - 6x + 9

= x² - 2 (x) (3) + 3²

= (x - 3)²

25x² - 30x + 9

= (5x)² - 2 (5x) (3) + 3²

= (5x - 3)²

64 –

= 8² – x²

= (8 – x) (8 + x)

8a² – 32b²

= 8 (a² – 4b²)

= 8 [ (a)² – (2b)² ]

= 8 (a – 2b) (a + 2b)

Questions

Solve: 3x + 6 = 2 (x – 10)

3x + 6 = 2x – 20

3x – 2x = – 20 – 6

x = – 26

x is an even number. Find the sum of next three odd numbers.

x , x + 1, x + 2, x + 3, x + 4, x + 5

Sum = x + 1 + x + 3 + x + 5

= 3x + 9

A square has a side of x cm. If its sides are increased by 2 cm each, what is the increase in its area?

Area before increase = (x) (x) = x²

Area after increase = (x + 2) (x + 2)

= x² + 2x + 2x + 4

= x² + 4x + 4

Increase in Area = x² + 4x + 4 – x²

= (4x + 4) cm²