# Factorize the polynomial a^3 + b ^3 + c^3 - 3abc

• Trail_Builder
In summary, the conversation discusses factoring the equation a^3 + b^3 + c^3 - 3abc and the attempts made to solve it. The solution that was found is (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc), but the conversation also mentions trying to factor it further and not being able to find a more elegant solution.
Trail_Builder

## Homework Statement

factorise: $$a^3 + b^3 + c^3 - 3abc$$

## The Attempt at a Solution

I had a few random attempts and found that $$(a + b + c)$$ is factor, and then dividing the original equation by (a + b + c) yeilds $$a^2 + b^2 + c^2 - ab - ac - bc$$

I can't figure out how to factorise this quadratic. I tried solving for A and it either doesn;t work, or I did it wrong lol. short of trying a zillion different factors (i'd rather not, I'd prefer a more elegant solution), what can I do?

thanks.

I don't think you can. I stuck it into both maple and mathematica and they return the same factors as you have found.

so I've factorised it as far as I can? :S

if so its a stupid problem lol. wouldve thought it would've given linear factors.

## 1. What is the purpose of factorizing a polynomial?

Factorizing a polynomial allows us to express it as a product of simpler terms, making it easier to solve and manipulate in mathematical operations.

## 2. What is the formula for factorizing the polynomial a^3 + b ^3 + c^3 - 3abc?

The formula for factorizing this polynomial is (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca).

## 3. How does factorizing the polynomial help in solving equations?

Factorizing allows us to break down a complex equation into smaller parts, making it easier to find the roots or solutions.

## 4. Can the polynomial a^3 + b ^3 + c^3 - 3abc be factored further?

No, this polynomial is already fully factored and cannot be reduced any further.

## 5. What is the significance of the -3abc term in the polynomial?

The -3abc term is significant because it ensures that the polynomial is divisible by (a + b + c), making the factorization process easier.

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