Question

find the value of x minus a whole cube + x minus b whole cube + x minus C whole cube is equal to 3 into x minus A into x minus b into x minus 3 then a + b + C is equal to 3 x​

Answer 1

Answer:

a+b+c = 3x.... Proved.

Step-by-step explanation:

We are given that

$(x-a)^{3}+(x-b)^{3}+(x-c)^{3}=3(x-a)(x-b)(x-c)$ ...... (1)

And, we have to prove that, $(a+b+c)=3x$...... (2)

Now we will prove it.

From equation (1), by simplification we get,

(x³-3x²a+3xa²-a³)+(x³-3x²b+3xb²-b³)+(x³-3x²c+3xc²-c³)=3(x-a)(x²-cx-bx+bc)

⇒(x³-3x²a+3xa²-a³)+(x³-3x²b+3xb²-b³)+(x³-3x²c+3xc²-c³)=3(x³-x²c-x²b+xbc-x²a+xac+xab-abc)

Cancelling equal terms from both sides we get,

⇒3x(a²+b²+c²)-(a³+b³+c³) =3x(ab+bc+ca)-3abc

⇒3x(a²+b²+c²-ab-bc-ca)= a³+b³+c³-3abc

We know the equation, a³+b³+c³-3abc= (a+b+c)(a²+b²+c²-ab-bc-ca)

⇒3x(a²+b²+c²-ab-bc-ca)= (a+b+c)(a²+b²+c²-ab-bc-ca)

⇒3x= a+b+c

⇒a+b+c = 3x

Hence proved.