Algebric identities homework

In summary, the given equation is a challenging olympic problem that requires factoring. By analyzing the equation, it is clear that it contains independent terms for a, b, and c. After expanding the equation, it can be simplified into ab(abc - a - b) + bc(abc - b - c) + ca(abc - c - a) + (a + b + c - 4abc). To factor this further, the factors (abc - a - b), (abc - b - c), and (abc - c - a) can be rewritten as (abc - a - b - c). There are no specific articles on factorization, but practicing with different problems can help improve skills in this
  • #1
Dinheiro
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Homework Statement


Factorate the expression a(1-b²)(1-c²) + b(1-c²)(1-a²) + c(1-a²)(1-b²) - 4abc

Homework Equations


Algebric identities

The Attempt at a Solution


It seems to be an olympic problem, and I can't find the right factors for it.
I would really appreciate the help and, if possible, could you indicate me some articles about factorizations, like this one, for mathematic olympiads.
Thanks
 
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  • #2
This was a bit hard to crack. I'll show you my thought chain. By the nature of the equation you see there are going to be independent a, b, c terms in the equation when expanded. This hints to a final factor containing constant term (±1) multiplied by another which contains independent terms (± a ± b ± c). After that I couldn't make any conclusions to decide the factors, and hence expanded the equation, which gives,

a²b²c + b²c²a + c²a²b - a²b - a²c - b²a - b²c - c²a - c²b + a + b + c - 4abc

Taking out ab, bc and ca as factors, and by cyclic symmetry in the equation,

ab(abc - a - b) + bc(abc - b - c) + ca(abc - c - a) + (a + b + c - 4abc)

Now if the factors (abc - a - b) , (abc - b - c) and (abc - c - a) could all be made of the form (abc - a - b - c) you can factor it out from all the terms. See if you can do this.

I'm not aware of any specific articles on factorization. Just play around with enough problems to give you an hint of what you can do in various situations.

PS: I'm curious to see if there are more elegant solutions to this from someone, than expanding out the whole equation.

Edit: Google gave these articles,
http://tutorial.math.lamar.edu/Classes/Alg/Factoring.aspx
http://www.qc.edu.hk/math/Resource/AL/Cyclic%20and%20symmetric%20polynomials.pdf
 
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What are algebric identities?

Algebric identities are mathematical equations or expressions that are always true for any values of the variables involved. These identities are used to simplify and solve complex algebraic problems.

Why is it important to learn about algebric identities?

Understanding algebric identities is crucial in solving various mathematical problems, especially in algebra. It helps in simplifying expressions, solving equations, and manipulating polynomials, which are essential skills in higher-level math and practical applications.

What are the common types of algebric identities?

The most commonly used algebric identities include the distributive property, commutative property, associative property, additive and multiplicative identities, and inverse properties. These identities help in solving problems involving addition, subtraction, multiplication, and division of variables.

How can I use algebric identities to solve homework problems?

To solve homework problems involving algebric identities, you must first identify the type of identity being used in the equation. Then, use the properties of that identity to simplify the expression and solve for the unknown variable.

Are there any tips for remembering algebric identities?

One helpful tip for remembering algebric identities is to practice regularly and use them in various math problems. Also, try to understand the logic behind each identity rather than just memorizing them. Additionally, make use of mnemonic devices or create your own visual aids to help you remember them better.

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