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

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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.
  • #1
Trail_Builder
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Homework Statement



factorise: [tex]a^3 + b^3 + c^3 - 3abc[/tex]

Homework Equations





The Attempt at a Solution



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

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.
 
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  • #2
I don't think you can. I stuck it into both maple and mathematica and they return the same factors as you have found.
 
  • #3
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.
 

Related to Factorize the polynomial a^3 + b ^3 + c^3 - 3abc

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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