USING THE FORMULA OF EXPANSION OF BINOMIAL OF POWER 3

Example 1 :

Find x³ - y³, if  x - y  =  5 and xy  =  14

Solution :

x³ - y³, if  x - y  =  5 and xy  =  14

(a³ - b³)  =  (a - b)³ + 3ab(a - b)

(x³ - y³)  =  (x - y)³ + 3xy(x - y)

By using the given values, we get

(x³ - y³)  =  5³ + 3(14)(5)

=  125 + 210

=  335

Example 2 :

If a + (1/a)  =  6, then find the value of a³ + 1/a³

Solution :

(a³ + b³)  =  (a + b)³ - 3ab(a + b)

a³ + (1/a)³  =   (a + (1/a))³ - 3a(1/a)(a + (1/a))

  =  6³ - 3(6)

  =  216 - 18

  =  198

Example 3 :

If x² + 1/x²  =  23, then find the value of x + (1/x) and x³ + (1/x³)

Solution :

a² + b²  =  (a + b)² - 2ab

x² + (1/x)²  =  (x + (1/x))² - 2x(1/x)

23  =  (x + (1/x))² - 2

23 + 2  =  (x + (1/x))²

 (x + (1/x))²  =  25

x + (1/x)  =  5

x³ + (1/x)³  =  (x + (1/x))³ - 3x(1/x)(x + (1/x))

  =  5³ - 3(5)

  =  125 - 15

  =  105

Hence the values of x + (1/x) and x³ + (1/x)³ are 5 and 105 respectively.

Example 4 :

If (y - (1/y))³  =  27, then find the value of y³ - (1/y)³

Solution :

(a³ - b³)  =  (a - b)³ - 3ab(a - b)

Given that :

(y - (1/y))³  =  27

(y - (1/y))³  =  3³

(y - (1/y))  =  3

(y³ - (1/y)³)  =  (y - (1/y))³ - 3y(1/y)(y - (1/y))

  =  27 - 3(3)

  =  27 - 9

  =  18

Hence the value of (y³ - (1/y)³)  is 18.

Example 5 :

Find the coefficient of the term a²b in the expansion of (3a + 2b)³.

Solution :

(3a + 2b)³

Using the formula for (a + b)³ = a³ + 3a² b + 3ab² + b³

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

= 27a³ + 3(9a²) (2b) + 3(3a)(4b²) + 8b³

= 27a³ + 54a²b + 36ab² + 8b³

So, the coefficient of a²b is 54.

Example 6 :

Factorise the following expression using formula.

m³ - n³

Solution :

a³  -b³ = (a - b) (a² + ab + b²)

m³ - n³ = (m - n) (m² + mn + n²)

Example 7 :

If 3x + 4y = 11 and xy = 2, find the value of 27x³ + 64y³

Solution :

3x + 4y = 11 and xy = 2

27x³ + 64y³ = (3x)³ + (4y)³

Here a = 3x and b = 4y

a³  + b³ = (a + b) (a² - ab + b²)

= (3x + 4y)((3x)² - (3x)(4y) + (4y)²)

= (3x + 4y)(9x² - 12xy + 16y²) -----(1)

(9x² + 16y²) = (3x)² + (4y)²

a² + b² = (a + b)² - 2ab 

= (3x + 4y)² - 2(3x)(4y)

= (3x + 4y)² - 24xy

= 11² - 24(2)

= 121 - 48

(9x² + 16y²)  = 73

By applying these values in (1), we get

= 11(73 - 12(2))

= 11(73 - 24)

= 11(49)

= 539

Example 8 :

If x² + 1/x² = 83, find the value of x³ - 1/x³

Solution :

x² + 1/x² = 83

a³ - b³ = (a - b) (a² + ab + b²)

a² + b² = (a + b)² - 2ab

x³ - 1/x³ = (x - 1/x) (x² + x(1/x) + (1/x)²)

= (x - 1/x) (x² + 1 + (1/x²)) -----(1)

x² + 1/x² = (x - 1/x)² + 2x(1/x)

= (x - 1/x)² - 2

Applying the value of x² + 1/x², we get

83 = (x - 1/x)² - 2

(x - 1/x)² = 83 + 2

= 85

(x - 1/x) = √85

By applying these values in (1), we get

=  √85 (83 + 1)

= 84√85

Example 9 :

Find the value of 27x³ + 64y³ + 36xy (3x + 4y) when x = 5 and y = -3

Solution :

27x³ + 64y³ + 36xy (3x + 4y)

Applying x = 5 and y = -3

= 27(5)³ + 64(-3)³ + 36(5)(-3) (3(5) + 4(-3))

= 27(125) + 64(-27) + 36(-15) (15 - 12)

= 3375 - 1728 + (-540) (3)

= 3375 - 1728 - 1620

= 3375 - 3348

= 27

Example 10 :

Find the following products :

(9m + 2n)(81m² - 18mn + 4n²)

Solution :

= (9m + 2n)(81m² - 18mn + 4n²)

= (9m + 2n)((9m)² - 9m(2n) + (2n)²)

a³ + b³ = (a + b) (a² - ab + b²)

= (9m)³ + (2n)³

= 729m³ + 8n³

Example 11 :

Find the following products :

(5 - 2x)(25 + 10x + 4x²)

Solution :

= (5 - 2x)(25 + 10x + 4x²)

= (5 - 2x)(5² + 5(2x) + 2²x²)

= (5 - 2x)(5² + 5(2x) + (2x)²)

= 5³ - (2x)³

= 125 - 8x³

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