A CUBE PLUS B CUBE FORMULA

On this webpage a cube plus 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

Question 1 :

Expand x³ + 2 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 "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

x³ + 2 ³   =  (x + 2) ((x)² - x(2) + 2 ²)

               =  (x + 2) (x² - 2x + 4)

Hence, the expansion of x³ + 2 ³ is(x + 2) (x² - 2x + 4) 


Question 2 :

Expand (3x)³ + 4 ³

Solution :

Here the question is in the form of (a ³ + b ³). Instead of a we have "3x" and instead of b we have "4" . Now we need to apply the formula (a+b)(a² - ab + b²) and we need to apply those values instead of a and b

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

               = (3x+4) ((3 ²x² - (3x)(4) + 4²)

               = (3x+4) (9x² - 12x + 16)

There fore the expansion of (3x)³ + 4 ³ is (3x+4) (9x² - 12x + 16)


Actually we have another formula for a³ + b³. We can derive this formula from (a + b)³

(a + b)³ = a³ + 3 a² b + 3 a b² + b³

            = a³ + b³+ 3 a b (a + b)

(a + b)³ - 3 a b (a + b) = a³ + b³

a³ + b³ = (a + b)³ - 3 a b (a + b)   a cube plus b cube formula

Question 3 :

Find x³ + y³ if x + y = 4 and x y = 5.

Solution :

Here the question is in the form of (a ³ + b ³).

a³ + b³ = (a + b)³ - 3 a b (a + b)

instead of a we have "x" and instead of b we have "y"

x³ + y³ = (x + y)³ - 3 x y (x + y)

             = 4³ - 3 (5) (4)

             = 64 - 60

             = 4

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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