FORMULA FOR A MINUS B THE WHOLE CUBE

Expand and simplify :

Problem 1 :

(x - 1)³

Solution :

(x - 1)³ is in the form of (a - b)³

Comparing (a - b)³ and (x - 1)³ , we get

a = x, b = 1

Substitute x for a and 1 for b

(x - 1)³ = x³ - 3(x²)(1) + 3(x)(1²) - 1³

(x - 1)³ = x³ - 3x² + 3x - 1

Problem 2 :

(x - 5)³

Solution :

(x - 5)³ is in the form of (a - b)³

Comparing (a - b)³ and (x - 5)³ , we get

a = x, b = 5

(x - 5)³ = x³ - 3(x²)(5) + 3(x)(5²) – (5)³

(x - 5)³ = x³ - 15x² + 75x - 125

Problem 3 :

(x - 4)³

Solution :

Here a = x and b = 4

(x - 4)³ = x³ - 3(x²)(4) + 3(x)(4²) – (4)³

(x - 4)³ = x³ - 12x² + 48x - 64

Problem 4 :

(x - y)³

Solution :

Here a = x and b = y

(x - y)³ = x³ - 3(x²)(y) + 3(x)(y²) – (y)³

(x - y)³ = x³ - 3x²y + 3xy² - y³

Problem 5 :

(2 - y)³

Solution :

Here a = 2 and b = y

(2 - y)³ = (2)³ - 3(2²)(y) + 3(2)(y²) – (y)³

(2 - y)³ = 8 - 12y + 6y² - y³

Problem 6 :

(2x - 1)³

Solution :

Here a = 2x and b = 1

(2x - 1)³ = (2x)³ - 3(2x)²(1) + 3(2x)(1²) - 1³

(2x - 1)³ = 8x³ - 12x² + 6x - 1

Problem 7 :

(3x - 1)³

Solution :

Here a = 3x and b = 1

(3x - 1)³ = (3x)³ - 3(3x)²(1) + 3(3x)(1²) - 1³

(3x - 1)³ = 27x³ - 27x² + 9x - 1

Problem 8 :

(2y – 3x)³

Solution :

Here a = 3x and b = 2y

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

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

Problem 9 :

Factorise x³ − x² − 𝑥 + 1

Solution :

x³ − x² − 𝑥 + 1

= x²(x - 1) - 1(x - 1)

= (x² - 1)(x - 1)

= (x² - 1²)(x - 1)

= (x + 1) (x - 1) (x - 1)

= (x + 1) (x - 1)²

Problem 10 :

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

Solution :

= 27x³ + 64y³ + 36xy(3x + 4y)

= 3³x³ + 4³y³ + 36xy(3x + 4y)

= (3x)³ + (4y)³ + 36xy(3x + 4y)

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a + b)³ = a³  + b³ + 3ab(a + b)

Here a = 3x and b = 4y

= (3x + 4y)³

Applying the given values of x and y, we get

= (3(5) + 4(-3))³

= (15 - 12)³

= 3³

= 27

So, the answer is 27.

Problem 11 :

Find the product of (9m + 2n) (81m² -  18mn + 4n²)

Solution :

= (9m + 2n) (81m² -  18mn + 4n²)

a³ + b³ = (a + b)(a² - ab + b²)

Here a = 9m and b = 2n

= (9m + 2n) (9² m² -  (9m)(2n) + 2² n²)

= (9m + 2n) [(9 m)² -  (9m)(2n) + (2 n)²]

= (9 m)³ + (2 n)³

= 729 m³ + 8 n³

Problem 12 :

Find the product of (3 + 5/x) (9 - 15/x + 25/x²)

Solution :

= (3 + 5/x) (9 - 15/x + 25/x²)

a³ + b³ = (a + b)(a² - ab + b²)

Here a = 3 and b = 5/x

= (3 + 5/x) (3² - 3(5/x) + (5/x)²)

= 3³ + (5/x)³

= 27 + 125/x³

Problem 13 :

Find the product of (5 - 2x) (25 + 10x + 4x²)

Solution :

= (5 - 2x) (25 + 10x + 4x²)

a³ - b³ = (a - b)(a² + ab + b²)

Here a = 5 and b = 2x

= (5 - 2x) (5² + 5(2x) + (2x)²)

= 5³ - (2x)³

= 125 - 8x³

Problem 14 :

Factroise (8/27)a³ + (4/9)a² - 2a/3 - 1

Solution :

= (8/27)a³ + (4/9)a² - 2a/3 - 1

= (2³/3³)a³ + ((2²/3²)a² - 2a/3 - 1

= ((2/3)a)³ + ((2/3)a)² - 2a/3 - 1

= [(2/3)a]² [(2/3)a + 1] - 1 [(2/3)a + 1]

= ([(2/3)a]² - 1) ((2/3)a + 1)

(2a/3 + 1)(2a/3 - 1) (2a/3 + 1)

= (2a/3 + 1)² (2a/3 - 1)

Problem 15 :

The value of

(0.1 x 0.1 x 0.1 + 0.02 x 0.02 x 0.02)/(0.2 x 0.2 x 0.2 + 0.04 x 0.04 x 0.04) is 

a)  0.0125    b) 0.125    c)  0.25    d)  0.5

Solution :

= (0.1 x 0.1 x 0.1 + 0.02 x 0.02 x 0.02)/(0.2 x 0.2 x 0.2 + 0.04 x 0.04 x 0.04)

= (0.1³ + 0.02³)/(0.2³ + 0.04³)

= (0.1³ + 0.02³)/(0.2³ (0.1³ + 0.02³))

= 1/0.2³

= 1/(2/10)³

= (10/2)³

= 1000/8

= 125

Problem 16 :

The value of

[(3/5)³ - (2/5)³] / [(3/5)² - (2/5)²]

Solution :

= [(3/5)³ - (2/5)³] / [(3/5)² - (2/5)²]

a³ - b³ = (a - b)(a² + ab + b²)

a² - b² = (a - b)(a + b)

= a³ - b³ / a² - b²

= (a - b) (a² + ab + b²) / (a + b)(a - b)

= (a² + ab + b²) / (a + b)

= ((3/5)² + (3/5)(2/5) + (2/5)²) / (3/5 + 2/5)

= ((9/25) + (6/25) + (4/25)) / ((3+2)/5)

= ((9 + 6 + 4)/25) / (5/5)

= 19/25