Yet another simple factorizing question

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In summary, the conversation is about factorizing a difference of two squares (a+1)^2-(a-1)^2 and the difference of two expressions that can be simplified to 4a. The main question is why the first form is used instead of the second, and it is explained that the first form is a factorization while the second is an expansion. The reason for using the first form is that it simplifies the process of factorization.
  • #1
alpha01
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[SOLVED] yet another simple factorizing question..

I won't re-write the full question, just one line on the numerator:

from the solutions, (a + 1)^2 − (a − 1)^2 factorizes to:

((a + 1) − (a − 1))((a + 1) + (a − 1))

I can see that this is just another form of:

(a + 1)(a + 1) - (a - 1)(a - 1)

but why is the former, and not the later used?

does it make it easier to go to the next step to complete factorization process?

If so please explain why.
 
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  • #2
alpha01 said:
does it make it easier to go to the next step to complete factorization process?
Well, it's going to depend on what the question asks next!
 
  • #3
the solution continues on like this:

= (a + 1 − a + 1)(a + 1 + a − 1)

= 4a

(the question is to factorize.. i don't know what you mean by "what does it ask next")
 
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  • #4
Ok, I think I get what you mean now. Well, your first expression is in the form x^2-y^2, which is a difference of two squares. We know that the factorisation of a difference of two squares is (x+y)(x-y); it just turns out that in this case the expression simplifies further.

The second expression you give in your first post is not a factorisation of (a+1)^2-(a-1)^2, but is an expansion.
 
  • #5
thank you, understood
 

Related to Yet another simple factorizing question

1. What is the purpose of factorizing in mathematics?

Factorizing is the process of breaking down a mathematical expression into smaller, simpler terms. This can help in solving equations, simplifying expressions, and understanding the relationships between different variables in an equation.

2. How do I factorize a quadratic equation?

To factorize a quadratic equation, you need to find two numbers that when multiplied together, equal the constant term and when added together, equal the coefficient of the middle term. These two numbers will be the factors of the quadratic equation.

3. Can all equations be factorized?

No, not all equations can be factorized. Only certain types of equations, such as quadratic equations, can be factorized using traditional methods. Other types of equations may require different techniques or may not be factorizable at all.

4. Why is factorizing important in algebra?

Factorizing is important in algebra because it allows us to simplify and solve equations, as well as identify patterns and relationships between different variables. It also helps in graphing and understanding the behavior of a function.

5. Are there any shortcuts or tricks for factorizing?

Yes, there are certain tricks and shortcuts that can be used for factorizing certain types of equations. For example, the difference of squares can be factored using the pattern (a+b)(a-b). However, it is important to understand the underlying concepts and principles of factorizing in order to use these shortcuts effectively.

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