Solve: (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = ?

Question

Examoogle · Accepted Answer

The correct answer is Option A: $a^3 + b^3 + c^3 - 3abc$. Substitute small integer values like $a=1, b=1, c=1$ into the expression to quickly evaluate and match with the options.

Step-by-Step Solution & Concepts:
Shortcut Trick: Substitute small integer values like $a=1, b=1, c=1$ into the expression to quickly evaluate and match with the options.

Formula Used: $$(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = a^3 + b^3 + c^3 - 3abc$$

Given: An algebraic product of a trinomial and a quadratic expression involving variables $a$, $b$, and $c$.

Steps: Identify the Algebraic Identity: Recognize that the given expression is a standard algebraic identity for the product of a trinomial and a second-degree polynomial.. Apply Expansion Formula: Multiplying each term yields the well-known expansion formula where all cross terms cancel out, leaving the sum of cubes minus $3abc$.. Verification via Substitution: Let $a=1, b=1, c=1$. The given expression evaluates to $(3)(1+1+1-1-1-1) = 3(0) = 0$. Option A evaluates to $1^3+1^3+1^3 - 3(1)(1)(1) = 3 - 3 = 0$.

Answer: A is correct because the product expands directly to the standard algebraic identity $a^3 + b^3 + c^3 - 3abc$.

Exam Tip: In competitive exams, variations of this identity often test the condition where $a+b+c=0$, which directly simplifies the equation to $a^3+b^3+c^3 = 3abc$.

Examoogle · Answer

Option B: $a^3 + b^3 + c^3 + 3abc$

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Option C: $(a+b+c)^3$

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Option D: $a^2 + b^2 + c^2$

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