Answer:
a³(b-c) + b³(c-a) + c³(a-b) = (a-b)(b-c)(c-a) (a + b + c)(a² + b² + c² - ab - bc - ca)
This is a factorized form of the polynomial, as you can see it factorizes as a combination of difference of squares and difference of cubes.
here is one way to factorize the polynomial a³(b-c) + b³(c-a) + c³(a-b) step by step:
First, we can factor out a³ from the first term, b³ from the second term, and c³ from the third term:
a³(b-c) + b³(c-a) + c³(a-b) = a³(b-c) + b³(c-a) + c³(a-b)
Next, we can factor out (b-c) from the first and second terms, and (c-a) from the second and third terms:
a³(b-c) + b³(c-a) + c³(a-b) = (b-c)(a³ + b³) + (c-a)(b³ + c³)
We can factor out (a-b) from the first and third terms, and then we can use difference of cubes to factorise out:
(b-c)(a³ + b³) + (c-a)(b³ + c³) = (b-c)(a-b)(a² + ab + b²) + (c-a)(a-b)(b² + ab + c²)
Now we can combine like terms, and we get:
(b-c)(a-b)(a² + ab + b²) + (c-a)(a-b)(b² + ab + c²) = (a-b)(b-c)(c-a)(a + b + c)(a² + b² + c² - ab - bc - ca)
The factorized form of the polynomial is (a-b)(b-c)(c-a) (a + b + c)(a² + b² + c² - ab - bc - ca)
This is a general formula for the factorization of a polynomial of the form a³(b-c) + b³(c-a) + c³(a-b)
By using difference of cubes we can factorise the above polynomial, hope it helps.