Question

use the rernainder theorem to find the factors of ( a-b )³ + (b-c )³ + ( c-a)³ 

Answer 1

We know that ( a-b )³ = a³ - 3a²b + 3 ab² - b³ ..........(i)

And if we put a - b = 0 c  a = b, and substitute this to the polynomial, we get 

f(x) = 0 + (a - c)³ + (c - a)³ = (a - c)³ - (a - c)³ = 0 

Hence, (a - b) is a factor. ⇒  a = b .... (ii)

Substiruong (1) in problem polynomial, we get 

f (x) = 0 + (b³ - 3b²c + 3bc² - c²) + (c³ - 3c²a + 3ca² - a³)

= - 3 b²c + 3 bc² - 3ca² + 3ca²  

= 3( -b²c + bc² - ca² + ca²)  

If we put b - c = 0 ⇒ b = c , and subsorute this ID the pdynorrual, we get: 

f (b = c) , 3 (-c² × c + c × c² - c × c² + c × c²) = 0

Hence, till new factors are 3 x (a - b) x (b - c) ... (iii)

Similarly if we had put c = a, we would have got similar result. 

So (c - a) is also a factor ..... (iv)

From (ii), (iu), and (iv), we get  

3(a - b)(b - c)(c - a) is a complete factorization of the oiven polynomial.