Maharashtra State BoardSSC (English Medium) 8th Standard

Expansion of (a + b)3

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Topics

Formula

  • (a + b)3 = a3 + 3a2b + 3ab2 + b3.

Notes

Expansion of (a + b)3:

(a + b)3 = (a + b)(a + b)(a + b) = (a + b)(a + b)2

(a + b)3 = (a + b)(a2 + 2ab + b2)

(a + b)3 = a(a2 + 2ab + b2) + b(a2 + 2ab + b2)

(a + b)3 = a3 + 2a2b + ab2 + ba2 + 2ab2 + b3

(a + b)3 = a3 + 3a2b + 3ab2 + b3

∴ (a + b)3 = a3 + 3a2b + 3ab2 + b3.

Example

Expand: (x + 3)3.

(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 xx x^2 xx 3 + 3 xx x xx (3)^2 + (3)^3`.

∴ `(x + 3)^3 = x^3 + 9x^2 + 27x + 27`.

Example

Expand: (3x + 4y)3.

(3x + 4y)3

`= (3x)^3 + 3(3x)^2(4y) + 3(3x)(4y)^2 + (4y)^3`.

`= 27x^3 + 3 xx 9x^2 xx 4y + 3 xx 3x xx 16y^2 + 64y^3`.

`= 27x^3 + 108x^2y + 144xy^2 + 64y^3`

Example

Expand: (41)3

(41)3

= (40 + 1)3

= (40)3 + 3 × (40)2 × 1 + 3 × 40 × (1)2 + (1)3

= 64000 + 4800 + 120 + 1

= 68921

Example

Expand: `((2m)/n + n/(2m))^3`.

`((2m)/n + n/(2m))^3`

`= ((2m)/n)^3 + 3((2m)/n)^2(n/(2m)) + 3((2m)/n)(n/(2m))^2 + (n/(2m))^3`

`= (8m^3)/(n^3) + 3((4m^2)/(n^2))(n/(2m)) + 3((2m)/n)(n^2/(4m^2)) + ((n^3)/8m^3)`

`= ((8m^3)/(n^3)) + ((6m)/n) + (3n)/(2m) + (n^3)/(8m^3)`

Example

Simplify.

(p + q)3 + (p - q)3

(p + q)3 + (p - q)3

= p3 + 3p2q + 3pq2 + q3 + p3 - 3p2q + 3pq2 - q3

= 2p3 + 6pq2

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