FIND THE CUBE OF A BINOMIAL

Problem 1 :

Find the cube of 2x + 3y.

Solution :

Write the formula for (a + b)³ = a³ + b³ + 3a²b + 3ab²

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

= 8x³ + 27y³ + 3(12x²y) + 3(18xy²)

(2x + 3y)³ = 8x³ + 27y³ + 36x²y + 54xy²

Problem 2 :

Find the cube of (2x - y)³

Solution :

Write the formula for (a - b)³ = a³ - b³ - 3a²b + 3ab²

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

= 8x³ - y³ - 3(4x²y) + 3(2xy²)

(2x - y)³ = 8x³ - y³ - 12x²y + 6xy²

Problem 3 :

If x = 37, what is the value of 8x³ + 72x² + 216x + 216?

Solution :

Instead of applying the values directly, let us simplify the given expression, we get

= 8x³ + 72x² + 216x + 216

= (2x)³ + 3(2x)² (6) + 3(2x) (6²)+ 6³

= (2x + 6)³

= (2(37) + 6)³

= (80)³

= 512000

Hence the value of 8x³ + 72x² + 216x + 216 is 512000.

Problem 4 :

If x – y = 8 and xy = 5, what is the value of x³ - y³ + 8(x + y)² ?

Solution :

Write the formula for a³ - b³ = (a - b)³ + 3ab(a - b)

x³ - y³ = (x - y)³ + 3xy(x - y)

x – y = 8 and xy = 5

Substitute the given values

x³ - y³ = (8)³ + 3(5) (8)

x³ - y³ = 512 + 120

x³ - y³ = 632

(x - y)² = x² + y² - 2xy

(8)² = x² + y² - 2(5)

64 = x² + y² - 10

x² + y² = 64 + 10

x² + y² = 74

Substitute x² + y² = 74 in this formula

(x + y)² = x² + y² + 2ab

       = 74 + 2(5)

      = 74 + 10

                                           (x + y)² = 84

x³ - y³ + 8(x + y)² = 632 + 8(84)

= 632 + 672

= 1304

Hence the value of x³ - y³ + 8(x + y)² is 1304.

Problem 5 :

If a – b = 5 and ab = 36, what is the value of a³ - b³?

Solution :

Write the formula for (a + b)² = a² + b² + 2ab

Given that, a – b = 5 and ab = 36

a – b = 5

Square both sides

(a - b)² = 5²

a² + b² - 2ab = 25

a² + b² - 2(36) = 25

a² + b² = 25 + 72

a² + b² = 97

a³ - b³ = (a - b) (a² + b² + ab)

Put ab = 36 and a² + b² = 97

a³ - b³ = 5(97 + 36)

a³ - b³ = 5(133)

a³ - b³ = 665

Hence the value of a³ - b³ is 665.

Problem 6 :

If a³ - b³ = 513 and a – b = 3, what is the value of ab?

Solution :

Write the formula for a³ - b³ = (a - b)³ + 3ab(a - b)

a³ - b³ = 513 and a – b = 3

(a - b)³ + 3ab(a - b) = 513

(3)³ + 3ab (3) = 513

27 + 9ab = 513

9ab = 513 – 27

9ab = 486

ab = 486 / 9

ab = 54

Hence the value of ab is 54.

Problem 7 :

If x = 19 and y = -12, find the value of 8x³ + 36x²y + 54xy² + 27y³.

Solution :

 8x³ + 36x²y + 54xy² + 27y³

Instead of applying the given values into the question directly, let us try to simplify the given expression first.

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

= (2x + 3y)³

By applying the given values of x and y, we get 

= (2x + 3y)³

= (2(19) + 3(-12))³

= (38 - 36)³

= 2³

= 8

Hence the value of 8x³ + 36x²y + 54xy² + 27y³ is 8.

Problem 8 :

If a = 15, what is the value of 8a³ + 60a² + 150a + 130?

Solution :

= 8(15)³ + 60(15)² + 150(15) + 130

= 8(3375) + 60(225) + 2250 + 130

= 27000 + 13500 + 2250 + 130

= 42880

Hence the value of 8a³ + 60a² + 150a + 130 is 42880.