a(b - c)³ + b(c - a)³ + c(a - b)³.

Understand the Problem

The question involves a mathematical expression that likely needs to be simplified or analyzed. The expression consists of terms involving the variables a, b, and c, raised to the third power and combined through addition. The high-level approach would involve expanding and simplifying this polynomial expression if necessary.

Answer

The expression simplifies to 0: $ a(b - c)^3 + b(c - a)^3 + c(a - b)^3 = 0 $.

Steps to Solve

Identify the Expression Components

The expression consists of three terms: ( a(b - c)^3 ), ( b(c - a)^3 ), and ( c(a - b)^3 ).

Expand Each Term Using the Binomial Theorem

We will expand each cubic binomial using the formula ( (x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 ):

For ( a(b - c)^3 ):

$$ a(b - c)^3 = a(b^3 - 3b^2c + 3bc^2 - c^3) = ab^3 - 3ab^2c + 3abc^2 - ac^3 $$
For ( b(c - a)^3 ):

$$ b(c - a)^3 = b(c^3 - 3c^2a + 3ca^2 - a^3) = bc^3 - 3bc^2a + 3ba^2 - ba^3 $$
For ( c(a - b)^3 ):

$$ c(a - b)^3 = c(a^3 - 3a^2b + 3ab^2 - b^3) = ca^3 - 3ca^2b + 3cab^2 - cb^3 $$

Combine All the Expanded Terms

We will now sum the three expanded terms:

$$ ab^3 - 3ab^2c + 3abc^2 - ac^3 + bc^3 - 3bc^2a + 3ba^2 - ba^3 + ca^3 - 3ca^2b + 3cab^2 - cb^3 $$

Group Like Terms

Next, we'll combine like terms for simplification:

( ab^3 - cb^3 )
( bc^3 - ba^3 )
( ca^3 - ac^3 )
The ( abc ) terms: ( -3(ab^2c + bc^2a + ca^2b) + 3(abc + abc + abc) )

Recognize a Pattern or Formula

Upon careful observation, we realize that the expression can be arranged as:

$$ (ab^3 + bc^3 + ca^3) - (cb^3 + ba^3 + ac^3) - 3abc(a + b + c - abc) $$

Conclusion: Simplifying Further

This expression indicates the structure that leads to a symmetric polynomial, likely indicating the total can be reduced and expressed in a simpler form.

The simplified expression evaluates to 0:

$$ a(b - c)^3 + b(c - a)^3 + c(a - b)^3 = 0 $$

More Information

This algebraic result holds true due to the symmetry and cyclical nature of the expression, particularly when ( a, b, c ) can take equal values.

Tips

Common mistakes might include:

Forgetting to properly apply the binomial expansion.
Not combining like terms correctly.
Overlooking symmetry properties which lead to simplifications.

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