Topics to be learn : Expansion of (x + a) (x +b) Expansion of (a + b)3 Expansion of (a - b)3 Expansion of (a + b + c)2

Recall :

(i) (a + b)² = a² + 2ab + b²,

Examples :

(a) (x + 2y)² = x² + 4xy + 4y²

(b) (101)² = (100 + 1)²

= (100)² + 2 x 100 x 1 + 1²

= 10000 + 200 +1 = 10201

(ii) (a - b)² = a²  - 2ab + b²,

Examples :

(a) (2x - 5y)² = 4x² - 10xy + 25y²

(b) (98)² = (100 - 2)²

= (100)² - 2 x 100 x 2 + (-2)²

= 10000 - 400 + 4 = 9604

(iii) (a + b) (a - b) = a²  - b²

Example :

(5m + 3n)(5m - 3n) = (5m)² – (3n)² = 25m² – 9n²

Activity : Expand (x + a)(x + b) using formulae for areas of a square and a rectangle.

(x + a)(x + b) = x² +ax + bx + ab = x² + (a +b)x + ab

Expansion of (x + a) (x +b) :

(x + a) and (x + b) are binomials with one term x in common, a and b are unequal terms.

Let us multiply them.

(x + a) (x + b) = x(x + b) + a(x + b)

= x² + bx + ax + ab

= x² + (b + a)x + ab

i.e. x² + (a + b)x + ab.

Examples :

(1) (x + 2)(x + 3) = x² + (2 + 3)x + (2  ×  3) = x² + 5x + 6

(2) (y + 4)(y - 3) = y² + (4 - 3)y + (4) ×  (-3) = y² + y - 12

(3) (x - 3)(x - 7) = x² + (-3 - 7)x + (-3)(-7) = x² - 10x + 21

Expansion of (a +b)³ :

(a + b)³ = (a + b)(a + b)(a + b)

= (a + b)(a + b)²

= (a + b)(a² + 2ab + b²)

= a(a² + 2ah + b²) + b(a² + 2ab + b²)

= a³ + 2a2b + ab² + a²b + 2ab² + b³

= a³ + 3a²b + 3ab² + b3.

(a + b) ³ = a³ + 3a²b + 3ab² + b³

Remember :

• The first and the fourth terms in the expansion are the cubes of the first and the second terms.
• The second term is three times the product of the first term squared and the second term.
• The third term is three times the product of the first term and the square of the second term.

Examples :

(1) (x + 3)³

We know that (a + b)³ = a³ + 3a²b + 3ab² + b³

In the given example, a = x and b = 3

∴ (x + 3)³ = (x) ³ + 3 × x² × 3 + 3 × x × (3)² + (3)³ = x³ + 9x² + 27x +27

(2) (3x + 4y)³ = (3x) ³ + 3(3x)²(4y) + 3(3x)(4y)² + (4y) ³

= 27x³ + 3 x 9 x² x 4y + 3 × 3x × 16y² + 64y³

= 27x³+ 108x²y + 144xy² + 64y³

Expansion of (a - b)³ :

∴ (a - b)³ = (a - b) (a - b) (a - b) = (a - b)(a - b)²

= (a - b)(a² - 2ab + b²)

= a(a² -2ab + b²) - b(a² - 2ab + b²)

= a³ - 2a²b + ab² - a²b + 2ab² - b³

= a³ - 3a²b + 3ab² - b³

∴ (a - b) ³ = a³ - 3a²b + 3ab² - b³

In words :

(first term - second term)³ =(first term)³ -3 x (first term)² x (second term) +3 x (first term) x (second term)² - (second term)³.

Examples :

(1) Expand (x - 2)³

(a - b) ³ = a³ - 3a²b + 3ab² - b³ Here taking , a = x and b = 2,

(x - 2)³ = (x) ³ - 3 × x² × 2 + 3 × x × (2)² - (2) ³

= x³ - 6x² + 12x - 8

(2) Find cube of 99 using the expansion formula.

(99)³ = (100 - 1)³ = (100)³ - 3 × (100)² × 1 + 3 × 100 × (1)² - 1³

= 1000000 - 30000 + 300 - 1 = 9,70,299

Expansion of (a + b + c)² :

(a + b + c)² = (a + b + c) × (a + b + c)

= a (a + b + c) + b (a + b + c) + c (a + b + c)

= a² + ab + ac + ab + b² + bc + ac + bc + c²

= a² + b² + c² + 2ab + 2bc + 2ac

∴ (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

Examples :

(1) Expand: (p + q + 3)²

= p² + q² + (3) ² + 2 × p × q + 2 × q × 3 + 2 × p × 3

= p² + q² + 9 + 2pq + 6q + 6p = p² + q² + 2pq + 6q + 6p + 9

(2) Fill in the boxes with appropriate terms in the steps of expansion.

(2p + 3m + 4n)²

= (2p)² + (3m)² + (4n)² + 2 × 2p × 3m + 2 × 3m × 4n + 2 × 2p × 4n

= 4p² + 9m² + 16n² + 12pm + 24mn + 16pn