Question

If a ^ 3 + (b - a) ^ 3 - b ^ 3 = k(a - b) then k =​

Answer 1

a^3+(b-a)^3 -b^3 = k(a-b)
We know that a^3 -b^3= (a-b) ( a^2+ ab + b^2)
So now we replace it in our equation
(b-a)^3 + (a-b)(a^2+ab+b^2) = k(a-b)
We take (a-b) as a common factor
-(a-b)^3+ (a-b)(a^2+ab +b^2) =k(a-b)
(a-b) (-(a-b)^2 + (a^2+ab + b^2) =k(a-b)
(a-b) ( - (a^2-2ab+b^2) + a^2 + ab + b^2) =k(a-b)
(a-b)(-a^2 +2ab -b^2 +a^2 +ab + b^2)=k(a-b)
(a-b)(2ab +ab)=k(a-b)
(a-b) 3ab =k(a-b)
SO k= 3ab

Answer 2

Explanation:

We can simplify the left-hand side of the equation using the identity for the sum of cubes:

a^3 + (b-a)^3 - b^3 = a^3 + (b^3 - 3b^2a + 3ba^2 - a^3) - b^3

= -3b^2a + 3ba^2

Now we can factor out the common factor of (a-b) from both sides of the equation:

a^3 + (b-a)^3 - b^3 = (a-b)(-3b^2 + 3ab) = k(a-b)

Canceling out the (a-b) factor from both sides of the equation, we get:

-3b^2 + 3ab = k

So the value of k is -3b^2 + 3ab.