A CUBE MINUS B CUBE FORMULA

On this webpage a cube minus b cube formula, that is (a³-b³) we are going to see some example problems based on this formula.

What is Algebraic identity?

An identity is an equality that remains true regardless of the values of any variables that appear within it.

Now let us see the

a cube minus b cube formula

Question 1 :

Expand ((5x)³ - 2 ³))

Solution:

Here the question is in the form of (a ³ - b ³). Instead of a we have "5x" and instead of b we have "2" . Now we need to apply the formula (a-b)(a² + ab + b²) and we need to apply those values instead of a and b

If we apply a cube minus b cube formula,we will get

(5x)³ - 2³  = (5x - 2) ((5x)² + 5x(2) + 2 ² )

                 = (5x-2) (5²x² + 5x(2) + 2²)

                 = (5x-2) (25x² + 10x + 2²)


Question 2 :

Expand (x³ - 1 ³)

Solution:

Here the question is in the form of (a ³ - b ³). Instead of a we have "x" and instead of b we have "1" . Now we need to apply the formula (a-b)(a² + ab + b²) and we need to apply those values instead of a and b

x³ - 1³  = (x - 1) (x² + x(1) + 1 ² )

             = (x-1) (x² + x + 1) 


Question 3 :

Expand (8x³ - 27 y³)

Solution:

            = (8x³ - 27 y³)

            = (2³x³ - 3³y³)

            = (2x)³ - (3y)³

Here the question is in the form of (a³ - b³). Instead of a we have "2x" and instead of b we have "3y" . Now we need to apply the formula (a-b)(a² + ab + b²) and we need to apply those values instead of a and b

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

                           = (2 x - 3 y) (2²x² + 6x y + 3²y²)

                           = (2 x - 3 y) (4 x² + 6x y + 9 y²)


Question 4 :

Expand (125x³ - 64 y³)

Solution:

            = (125 x³ - 64 y³)

            = (5³x³ - 4³y³)

            = (5x)³ - (4y)³

Here the question is in the form of (a³ - b³). Instead of a we have "5x" and instead of b we have "4y" . Now we need to apply the formula (a-b)(a² + ab + b²) and we need to apply those values instead of a and b

(5x)³ - (4y)³  = (5 x - 4 y) ((5x)² + 5x (4y) + (4y)²)

                           = (5 x - 4 y) (25 x² + 20 x y + 16 y²)

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(a + b)² = a² + 2 ab + b²

(a - b)² = a² - 2 ab + b²

a² - b² = (a + b) (a - b)

(x+a)(x+b)=x²+(a+b)x+ab

(a+b)³=a³+3a²b+3ab²+b³

(a-b)³=a³-3a²b+3ab²-b³

(a³+b³)= (a+b)(a²-ab+b²)

(a³-b³)=(a-b)(a²+ab+ b²)

(a+b+c)²= a²+b²+c²+2ab+2bc+2ca